Polygon

Polygon: A Fundamental Plane Figure Bounded by Line Segments

For other uses, see Polygon (disambiguation) .

Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.

In the realm of geometry , where forms and spaces are meticulously defined, a polygon stands as a foundational concept: a plane figure constructed entirely from a finite sequence of straight line segments . These segments are not haphazardly arranged; they are connected end-to-end to elegantly form a closed polygonal chain . This closure is paramount, distinguishing a polygon from a mere sequence of lines. To exist as a polygon, the figure must completely enclose a region, returning to its starting point without ambiguity.

The individual straight line segments that constitute this closed polygonal chain are formally recognized as its edges , or more colloquially, its sides. The precise points where any two consecutive edges converge and meet are termed the polygon’s vertices or corners. The nomenclature for polygons is often straightforward, determined by the count of these sides: an n-gon denotes a polygon possessing n sides. For instance, a triangle , the simplest Euclidean polygon, is a 3-gon, while a quadrilateral is a 4-gon, and so on.

A particularly well-behaved category of polygons is the simple polygon . This designation applies to any polygon that, in its construction, does not intersect itself. To be more precise, the only permissible intersections among the line segments forming the polygon are the shared endpoints of consecutive segments within the polygonal chain. When one speaks of a simple polygon , one is often referring to the boundary that encloses a distinct region of the plane . This enclosed region itself is referred to as a solid polygon, or its interior, known as its body, a polygonal region, or polygonal area. In practical contexts, particularly when the discussion implicitly concerns only these simple and solid forms, the term “polygon” frequently serves as a shorthand for either a simple polygon (the boundary) or a solid polygon (the enclosed region). Such precision is often lost on the casual observer, but its importance is fundamental.

However, not all polygons adhere to such pristine simplicity. A polygonal chain possesses the geometric freedom to cross over itself, giving rise to more intricate forms such as star polygons and other varieties of self-intersecting polygons . These figures, while still defined by connected line segments and closure, introduce complexities in how their “interior” is perceived and measured. Furthermore, some mathematical frameworks extend the definition of a polygon beyond the confines of a single plane . In these contexts, closed polygonal chains existing within Euclidean space (or even higher-dimensional spaces) are considered a type of polygon, specifically a skew polygon , even when their vertices do not all reside on a common plane . This broader perspective acknowledges the form’s inherent structure regardless of its spatial embedding.

Ultimately, a polygon can be understood as a 2-dimensional manifestation of the more encompassing concept of a polytope , which generalizes this idea to any number of dimensions. Just as a polygon defines a boundary and an interior in two dimensions, a polytope does so in n dimensions. The elegance of mathematics, it seems, is in its recursive definitions. And, as is often the case with fundamental ideas, numerous other generalizations of polygons have been meticulously defined over time, each serving specific, often esoteric, purposes.

Etymology

The very word “polygon” carries its meaning within its ancient roots, a testament to the straightforwardness of early geometric observation. It descends from the Greek adjective πολύς (pronounced polús), meaning ‘much’ or ‘many’, combined with γωνία (pronounced gōnía), which translates to ‘corner’ or ‘angle’. Thus, a polygon is, quite literally, a “many-cornered” or “many-angled” figure. A perfectly descriptive, if rather uninspired, name. Interestingly, some etymological analyses have also posited that γόνυ (pronounced gónu), meaning ‘knee’, might be a potential origin for the ‘gon’ component, perhaps alluding to the bending or articulation at a polygon’s vertices . While a charming thought, it hardly changes the fundamental truth of its definition.

Classification

Polygons, like all things in a structured universe, can be categorized, their properties dissected and assigned to neatly labeled boxes. This classification allows for a more rigorous understanding of their diverse forms, moving beyond merely counting their sides.

Some different types of polygon

Number of sides

The most immediate and fundamental method of classifying polygons is, predictably, by the sheer quantity of their sides. An n-gon is simply a polygon with n sides, a system that, while lacking in poetic flourish, is undeniably effective. This count directly dictates the number of vertices and, in most cases, the fundamental geometric properties of the figure.

Convexity and intersection

Beyond the simple count of sides, polygons are frequently characterized by their overall shape, specifically concerning their convexity or the particular nature of their non-convexity. This defines how the figure “bends” or “folds” in on itself.

• Convex : A polygon is deemed convex if, and only if, any straight line drawn completely through its interior (and not merely tangent to an edge or vertex ) will invariably intersect its boundary at precisely two points. This property has several direct consequences: all of its interior angles must be strictly less than 180°, and, perhaps more intuitively, any line segment connecting two points on the polygon’s boundary will lie entirely within or on the boundary of the polygon itself. There are no “indentations” or “caves” in a convex polygon . This condition holds true for polygons irrespective of the underlying geometry, not exclusively in Euclidean geometry . It represents a robust and well-behaved form, the geometric ideal, if you will.

• Non-convex: As the name rather plainly suggests, a non-convex polygon is simply one that fails the criteria for convexity . This means that at least one line can be found that intersects its boundary more than twice, or, equivalently, a line segment can be drawn between two points on its boundary that, at some point, passes outside the polygon’s defined interior. These are the polygons with quirks, deviations from the perfectly rounded ideal.

• Simple : A simple polygon is characterized by a boundary that does not intersect itself at any point other than its vertices . All convex polygons are, by definition, simple. This category represents the most commonly visualized and studied polygons, forming a single, unbroken loop.

• Concave : This is a specific subset of non-convex polygons, distinguished by being both non-convex and simple . The defining feature of a concave polygon is the presence of at least one interior angle that measures greater than 180°. These are the polygons that appear to have been “pushed in” or “dented” from the outside.

• Star-shaped : A polygon is considered star-shaped if its entire interior is visible from at least one internal point, without any of the polygon’s edges obstructing the view. This implies that the polygon must be simple , but it can be either convex or concave . All convex polygons are inherently star-shaped , as every point within a convex polygon can “see” the entire interior.

• Self-intersecting : These polygons venture into more complex territory, where the boundary itself crosses over itself. The term “complex” is occasionally employed as a contrast to “simple” in this context, but such usage risks creating confusion with the more specialized concept of a complex polygon , which exists in the abstract realm of the complex Hilbert plane and involves two complex dimensions, a notion far removed from the simple visual intersection. Such semantic nuances are, of course, a constant joy.

• Star polygon : A star polygon is a specific type of self-intersecting polygon that exhibits a regular, symmetrical pattern of self-intersection. It’s a common misconception, or perhaps just wishful thinking, that a polygon can be both a star and star-shaped . These are distinct properties; a star polygon self-intersects, while a star-shaped polygon is simple but may be concave .

Equality and symmetry

Further refinements in classification arise from the inherent symmetries and equalities within a polygon’s structure, elevating some forms above the mundane.

• Equiangular : In an equiangular polygon , every single corner angle possesses the exact same measure. A pleasing uniformity, to be sure.

• Equilateral : An equilateral polygon is defined by having all of its edges (or sides) of precisely the same length. A balance of dimensions.

• Regular : The paragon of polygonal forms, a regular polygon is one that is simultaneously both equilateral and equiangular . These are the perfectly balanced, aesthetically pleasing shapes that often come to mind when one first considers polygons.

• Cyclic : A cyclic polygon is one whose vertices all lie precisely upon a single circle . This encompassing circle is known as the circumcircle of the polygon.

• Tangential : Conversely, a tangential polygon is defined by the property that all of its sides are tangent to a single inscribed circle within its interior.

• Isogonal or vertex-transitive : In an isogonal polygon, all of its corners are indistinguishable under the polygon’s symmetry operations ; they lie within the same symmetry orbit . Such a polygon is necessarily both cyclic and equiangular .

• Isotoxal or edge-transitive : An isotoxal polygon is one where all of its sides are congruent and lie within the same symmetry orbit . This type of polygon is consequently both equilateral and tangential .

The property of regularity, that coveted state of geometric perfection, can be expressed through these symmetry properties as well: a polygon achieves regularity if and only if it is simultaneously both isogonal and isotoxal. Equivalently, it is regular if it is both cyclic and equilateral . A curious exception to the usual visual order is the non-convex regular polygon, which is famously referred to as a regular star polygon , maintaining its internal symmetries despite its self-intersections.

Miscellaneous

Beyond these primary categorizations, other properties allow for further, more specialized descriptions of polygons.

• Rectilinear : A rectilinear polygon is defined by the rather rigid constraint that all of its sides meet at precise right angles. This means every one of its interior angles will be either 90° or 270°. These are the polygons that adhere strictly to a grid, devoid of any sloping lines.

• Monotone with respect to a given line L: This is a more topological classification. A polygon is monotone with respect to a given line L if every line orthogonal (perpendicular) to L intersects the polygon’s boundary no more than twice. This essentially means the polygon “flows” in a consistent direction relative to L, without doubling back on itself excessively.

Properties and formulas

For the sake of clarity and consistency, and perhaps a slight aversion to unnecessary complications, Euclidean geometry is assumed throughout the following discussion. Deviations from this standard are, as always, noted.

Partitioning an n -gon into n − 2 triangles

Angles

Every polygon, by its very definition, possesses an equal number of corners and sides. Each of these corners is associated with several angles, the two most commonly discussed being the interior and exterior angles.

• Interior angle – The sum of the interior angles of any simple n-gon (a polygon with n sides) is invariably given by the formula ( n − 2) × π radians , which translates to ( n − 2) × 180 degrees . This fundamental property arises from the elegant observation that any simple n-gon can be decomposed, or triangulated, into precisely (n − 2) triangles , and each triangle itself possesses an angle sum of π radians or 180 degrees . It’s a neat trick, if you appreciate such things. For a convex regular n-gon , where all angles are equal, the measure of any single interior angle can be calculated as:

  $$ \left(1-{\tfrac {2}{n}}\right)\pi $$ radians or $$ 180-{\tfrac {360}{n}} $$ degrees .

  The interior angles of regular star polygons —those fascinating, self-intersecting forms—were first systematically investigated by Louis Poinsot . In the very paper where he meticulously described the four regular star polyhedra , he provided the formula for these angles. For a regular $$ {\tfrac {p}{q}} $$ -gon (which is a p-gon with a central density of q), each interior angle is given by: $$ {\tfrac {\pi (p-2q)}{p}} $$ radians or $$ {\tfrac {180(p-2q)}{p}} $$ degrees . It’s a formula that hints at the underlying order even in seemingly chaotic forms.

• Exterior angle – The exterior angle at a vertex is, by definition, the supplementary angle to the interior angle at that same vertex . Consider the simple act of tracing the perimeter of a convex n-gon : as you navigate each corner , you “turn” by a specific amount. This amount is precisely the exterior angle . If you complete a full circuit around the entire polygon, you will have executed one full turn . Consequently, the sum of all the exterior angles of any convex polygon must always be 360°. This elegant principle can be extended to include concave simple polygons , provided that external angles corresponding to turns in the opposite direction are subtracted from the cumulative total. When tracing around an n-gon in its most general sense, encompassing even self-intersecting polygons , the sum of the exterior angles (representing the total angular rotation accumulated at the vertices ) can be any integer multiple d of 360°. Here, d is a significant topological invariant known as the density or turning number of the polygon. For example, a pentagram has a turning number of 2, resulting in an exterior angle sum of 720°, while a degenerate angular “eight” or antiparallelogram can have a turning number of 0.

Area

Coordinates of a non-convex pentagon

Calculating the area enclosed by a polygon is a fundamental geometric problem, one that becomes particularly interesting when dealing with irregular or complex shapes. In this section, we assume the vertices of the polygon are ordered sequentially as $$ (x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1}) $$ . For the sake of mathematical convenience in certain formulas, it is customary to denote $$ (x_{n},y_{n}) = (x_{0},y_{0}) $$ , effectively closing the loop.

Simple polygons

• Further information: Shoelace formula

If the polygon in question is non-self-intersecting—that is, a simple polygon —its signed area A can be elegantly computed using the following formula:

$$ A={\frac {1}{2}}\sum {i=0}^{n-1}(x{i}y_{i+1}-x_{i+1}y_{i})\quad {\text{where }}x_{n}=x_{0}{\text{ and }}y_{n}=y_{0}, $$

Alternatively, and perhaps with a touch more algebraic flair, one can employ determinants :

$$ 16A^{2}=\sum {i=0}^{n-1}\sum {j=0}^{n-1}{\begin{vmatrix}Q{i,j}&Q{i,j+1}\Q_{i+1,j}&Q_{i+1,j+1}\end{vmatrix}}, $$

where $$ Q_{i,j} $$ represents the squared Euclidean distance between vertex $$ (x_{i},y_{i}) $$ and vertex $$ (x_{j},y_{j}) $$ . This particular formulation, while less common, highlights deeper mathematical connections.

The resultant signed area is contingent upon the specific ordering of the vertices and the chosen orientation of the plane . Conventionally, the positive orientation is established by a counterclockwise rotation that maps the positive x-axis to the positive y-axis. If the vertices are listed in a counterclockwise sequence (i.e., according to this positive orientation ), the signed area will be positive. Conversely, if the vertices are ordered clockwise, the signed area will be negative. In either scenario, the absolute value of the result provides the true geometric area . This versatile formula is widely known as the shoelace formula or the surveyor’s formula, a moniker derived from the crisscrossing pattern of its terms, reminiscent of tying a shoelace. It’s a rather practical piece of mathematics, surprisingly useful for something so abstract.

The area A of a simple polygon can also be determined if one possesses knowledge of the lengths of its sides, denoted as a₁, a₂, …, a_n, along with its exterior angles , θ₁, θ₂, …, θ_n. The formula, though more elaborate, offers an alternative perspective:

$$ {\begin{aligned}A={\frac {1}{2}}(a_{1}[a_{2}\sin(\theta {1})+a{3}\sin(\theta _{1}+\theta {2})+\cdots +a{n-1}\sin(\theta _{1}+\theta {2}+\cdots +\theta {n-2})]\{}+a{2}[a{3}\sin(\theta {2})+a{4}\sin(\theta _{2}+\theta {3})+\cdots +a{n-1}\sin(\theta {2}+\cdots +\theta {n-2})]\{}+\cdots +a{n-2}[a{n-1}\sin(\theta _{n-2})]).\end{aligned}}} $$

This formula was meticulously described by A.M. Lopshits in 1963, adding another tool to the geometrician’s arsenal.

For polygons that are specifically drawn on an equally spaced grid, where all of their vertices perfectly align with grid points, a delightfully simple and elegant formula emerges: Pick’s theorem . This theorem states that the polygon’s area can be calculated directly from the number of interior grid points (I) and boundary grid points (B): A = I + B/2 − 1. It’s a charmingly direct way to quantify space for those who appreciate a well-defined lattice.

Furthermore, for every polygon with a perimeter p and an area A, the fundamental isoperimetric inequality holds true:

$$ p^{2}>4\pi A $$

This inequality reveals a profound relationship, demonstrating that among all shapes with a given perimeter, the circle encloses the maximum area . All other polygons, by their very nature, are less efficient in this regard.

In a more abstract but equally fascinating vein, the Bolyai–Gerwien theorem asserts that for any two simple polygons that possess identical area , the first polygon can always be meticulously dissected into a finite number of polygonal pieces. These pieces can then be rearranged, without any overlap or gaps, to perfectly form the second polygon. This theorem speaks to a deep equivalence between shapes that share the same measure of space, a testament to the transformative power of dissection.

It is a common oversight, often leading to erroneous assumptions, that the lengths of a polygon’s sides are sufficient to determine its area . In general, this is not the case; a polygon can often be “flexed” or reshaped while maintaining side lengths, thus altering its enclosed area . However, if a polygon is both simple and cyclic —meaning its vertices lie on a circle —then, and only then, do its side lengths uniquely define its area . Among all n-gons with a specified set of side lengths, the one that encloses the greatest area is always the cyclic one. Moreover, for all n-gons sharing a given perimeter, the polygon that achieves the maximum area is the regular polygon (which, by definition, is also cyclic ). It seems regularity, once again, leads to optimal efficiency.

Regular polygons

Many specialized and often more concise formulas are applicable specifically to the areas of regular polygons , simplifying calculations for these ideal forms.

The area A of a regular polygon can be expressed elegantly in terms of r, the radius of its inscribed circle (the largest circle that can fit entirely within the polygon and be tangent to all its sides), and p, its perimeter:

$$ A={\tfrac {1}{2}}\cdot p\cdot r. $$

This radius r is also frequently referred to as the polygon’s apothem and is often denoted by the variable a.

Alternatively, the area A of a regular n-gon can be calculated using R, the radius of its circumscribed circle (the unique circle that passes through all of the vertices of the regular n-gon ):

$$ A=R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=R^{2}\cdot n\cdot \sin {\frac {\pi }{n}}\cdot \cos {\frac {\pi }{n}} $$

These formulas highlight the deep connection between regular polygons and circles . Indeed, as the number of sides n of a regular polygon approaches infinity, the polygon itself approaches the form of a circle , and its area formula converges to the familiar area of a circle with radius R: $$ \lim _{n\to +\infty }R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=\pi \cdot R^{2} $$ . A neat demonstration of limits, if you’re into that sort of thing.

Self-intersecting

The concept of area becomes significantly more ambiguous and requires careful definition when dealing with self-intersecting polygons . The very act of the boundary crossing itself complicates what constitutes “inside” and “outside,” leading to at least two distinct interpretations, each yielding a different result.

• One definition employs the same formulas used for simple polygons , but with an important modification: particular regions within the polygon are assigned a “density” factor, which effectively multiplies their contribution to the total area . For example, the central convex pentagon that forms the core of a pentagram is considered to have a density of 2, meaning its area is counted twice. In the case of a cross-quadrilateral (a figure resembling an “8”), the two triangular regions formed by the self-intersection are assigned opposite-signed densities. When their areas are summed, this can result in a total area of zero for the entire figure, which, while mathematically consistent, can be counter-intuitive.

• A second approach considers the enclosed regions as actual point sets. Under this definition, the area is simply the measure of the total plane covered by the polygon, regardless of how many times the boundary crosses over itself. This corresponds to calculating the area of one or more simple polygons that share the exact same outline as the self-intersecting polygon , effectively ignoring the self-intersections for the purpose of defining the boundary. For the cross-quadrilateral, this means it would be treated as two distinct simple triangles , and their absolute areas would be summed. This method often aligns more closely with intuitive notions of “space occupied.”

Centroid

The centroid of a polygon represents its geometric center, a point of balance. As with area , its calculation depends on whether one is considering the solid polygon or merely its set of vertices . Using the same convention for vertex coordinates as established in the preceding section, the coordinates of the centroid $$ (C_x, C_y) $$ of a solid simple polygon are given by:

$$ C_{x}={\frac {1}{6A}}\sum {i=0}^{n-1}(x{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}), $$

$$ C_{y}={\frac {1}{6A}}\sum {i=0}^{n-1}(y{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}). $$

It is crucial to note that in these formulas, the signed value of the area $$ A $$ (as obtained from the shoelace formula ) must be utilized. Failure to do so will, predictably, yield incorrect results.

For triangles (n = 3), a unique simplification occurs: the centroid of the vertices and the centroid of the solid triangular shape coincide perfectly. However, this convenient equivalence is generally not true for polygons with n > 3 sides. For polygons with more than three sides, the two centroids are distinct. The centroid of the vertex set of a polygon with n vertices is simply the average of their coordinates:

$$ c_{x}={\frac {1}{n}}\sum {i=0}^{n-1}x{i}, $$

$$ c_{y}={\frac {1}{n}}\sum {i=0}^{n-1}y{i}. $$

This distinction between the center of the boundary points and the center of the enclosed mass is a subtle but important one, often overlooked by those who prefer simple answers.

Generalizations

The concept of a polygon, while seemingly straightforward in its Euclidean plane form, has been ingeniously expanded and generalized in numerous ways by mathematicians, pushing the boundaries of what a “many-sided figure” can entail. Because, apparently, the simple reality wasn’t enough.

• A spherical polygon ventures beyond the flat plane , existing instead on the curved surface of a sphere . It is defined as a circuit of arcs of great circles (which serve as its sides) connecting a sequence of vertices that lie on the sphere’s surface . This generalization is particularly notable because it permits the existence of the digon , a polygon with only two sides and two corners, a configuration utterly impossible in a flat Euclidean plane . Spherical polygons are not mere curiosities; they play a critical role in fields such as cartography (the art and science of map making), where the Earth’s curved surface necessitates such geometric adaptations, and in Wythoff’s construction of the uniform polyhedra .

• A skew polygon is a polygon that refuses to be confined to a single plane . Instead, its vertices and edges zigzag through three (or even more) dimensions. While its edges are still straight line segments and it forms a closed polygonal chain , the entire figure does not lie flat. The Petrie polygons of the regular polytopes (higher-dimensional analogues of polygons and polyhedra ) are well-known and visually striking examples of skew polygons .

• An apeirogon , from the Greek “apeiros” meaning infinite, is a degenerate polygon of infinitely many sides. It represents an infinite sequence of sides and angles, which, by its very nature, is not closed in the traditional sense but also possesses no ends, extending indefinitely in both directions. It’s what happens when you decide “many” isn’t quite enough.

• A skew apeirogon takes the concept of the apeirogon and lifts it out of the plane . It is an infinite sequence of sides and angles that, like its finite skew polygon counterpart, does not lie within a single flat plane .

• A polygon with holes describes a region of the plane that is connected (meaning you can travel between any two points within it without leaving the region) but is “multiply-connected” topologically. It possesses one external boundary, defining its overall shape, and one or more interior boundaries, which are the “holes” within it. Imagine a doughnut, but flat and with straight sides.

• A complex polygon is a more abstract and algebraic generalization. It is a configuration that is analogous to an ordinary polygon but exists within the complex plane —a mathematical space composed of two real dimensions and two imaginary dimensions. This moves polygons far beyond simple visualization and into the realm of abstract algebra and higher-dimensional geometry.

• An abstract polygon strips away all geometric properties, focusing purely on the combinatorial relationships between its elements. It is an algebraic partially ordered set that meticulously represents the various constituent elements of a polygon (such as its sides and vertices ) and their intricate connectivity. A “real” geometric polygon, the kind one can draw or visualize, is then considered a “realization” of its associated abstract polygon . Depending on the specific mapping or “realization,” all the geometric generalizations described here can be understood as manifestations of their underlying abstract polygon structures.

• A polyhedron extends the concept of a polygon into three dimensions. It is a three-dimensional solid bounded by flat polygonal faces, acting as the direct three-dimensional analogue to a polygon in two dimensions. These faces are themselves polygons. The corresponding shapes in four or higher dimensions are generally referred to as polytopes . (It’s worth noting that in some alternative conventions, the terms “polyhedron ” and “polytope ” are used more broadly to refer to such figures in any dimension, with the primary distinction often being that a polytope is necessarily bounded, while a polyhedron might not be.)

Naming

The convention for naming polygons, much like their definition, is rooted in Late Latin and Greek . The word polygōnum (a noun in Late Latin ) stems from the Greek πολύγωνον (polygōnon/polugōnon), which, when used as a noun, is the neuter form of the masculine adjective πολύγωνος (polygōnos/polugōnos), meaning “many-angled.” A rather literal lineage.

Individual polygons are typically named, and sometimes classified, based on the number of sides they possess. This involves combining a Greek -derived numerical prefix with the suffix “-gon.” For example, a 5-sided polygon is a pentagon , and a 12-sided polygon is a dodecagon . However, there are a few notable exceptions to this otherwise consistent system: the triangle (3 sides), quadrilateral (4 sides), and nonagon (9 sides) employ Latin or unique English roots for their common names, though their Greek-derived alternatives (trigon, tetragon, enneagon, respectively) are also recognized, if less frequently used.

Beyond the relatively common decagons (10-sided) and dodecagons (12-sided), mathematicians, perhaps weary of inventing increasingly convoluted Greek-Latin hybrid names, generally resort to a simpler, more pragmatic numerical notation. Thus, one typically refers to a 17-gon or a 257-gon, a clear indication that practicality often trumps linguistic elegance in higher mathematics.

Nevertheless, exceptions persist for side counts that are readily expressed in verbal form (such as “20-gon” for icosagon or “30-gon” for triacontagon ), particularly when used by non-mathematicians who might find “icosagon” less intuitive. Furthermore, some special polygons have earned their own distinct and memorable names. A prime example is the regular star pentagon , which is more widely recognized as a pentagram or pentacle, a name imbued with cultural and historical significance far beyond its geometric definition.

Polygon names and miscellaneous properties

| Name The user needs an extended and detailed Wikipedia article about Polygons, written in Emma’s style.

Okay, another one. A polygon. The rudimentary geometric forms that are supposed to capture the essence of ‘many-sided’. As if the universe needed another straightforward definition. Fine. Let’s make it interesting, or at least, less offensively dull than it might otherwise be.

Polygon: A Fundamental Plane Figure Bounded by Line Segments

For other uses, see Polygon (disambiguation) .

Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.

In the elegant, often unforgiving, discipline of geometry , a polygon presents itself as a foundational concept: a plane figure meticulously constructed from a finite sequence of straight line segments . These segments are not merely scattered components; they are precisely connected end-to-end, forming what is known as a closed polygonal chain . This closure is not a mere suggestion; it is a critical, immutable requirement, distinguishing a polygon—a true enclosed form—from a simple, open sequence of lines that merely wanders without resolution. To genuinely qualify as a polygon, the figure must unequivocally enclose a distinct region, its path returning with precision to its point of origin.

The individual, rectilinear line segments that collectively compose this closed polygonal chain are formally designated as its edges , or, in less formal discourse, its sides. The exact points in space where any two consecutive edges converge and meet are precisely termed the polygon’s vertices or corners. The system of nomenclature for polygons is, thankfully, often quite direct, primarily dictated by the sheer count of these sides: an n-gon denotes a polygon possessing exactly n sides. For illustrative purposes, a triangle , which one might consider the irreducible minimum for a Euclidean polygon, is a 3-gon. Following this logical progression, a quadrilateral is a 4-gon, a pentagon a 5-gon, and so forth, a system that, while perhaps lacking in poetic grandeur, is undeniably clear and efficient.

Among the various manifestations of polygons, the simple polygon holds a place of particular prominence, largely due to its predictable and well-behaved nature. This classification is assigned to any polygon whose boundary, in its intricate construction, does not intersect itself at any point other than the shared endpoints of consecutive segments within the polygonal chain. All convex polygons , for instance, are inherently simple. When the term “simple polygon ” is invoked, it commonly refers to the demarcation, the boundary, that elegantly encloses a single, distinct region of the plane . This enclosed region itself is often referred to as a solid polygon. Its internal expanse, the space it occupies, is known as its body, or more explicitly, a polygonal region or polygonal area. In practical, everyday contexts, especially when discussions implicitly focus on these uncomplicated, solid forms, the term “polygon” frequently serves as a concise shorthand, referring interchangeably to either the simple polygon (the boundary itself) or the solid polygon (the filled interior). This subtle but critical distinction between the boundary and the space it defines, while often glossed over in casual conversation, is fundamentally important for precise geometric understanding.

However, the world of polygons is not solely populated by such pristine, non-intersecting forms. A polygonal chain possesses the inherent geometric freedom to cross over itself, thereby giving rise to more intricate and topologically complex figures such as star polygons and a broader array of self-intersecting polygons . These figures, while faithfully adhering to the core definition of connected line segments and ultimate closure, introduce significant complexities in how their “interior” is conceptually defined and quantitatively measured. Furthermore, certain advanced mathematical frameworks extend the very definition of a polygon beyond the strict confines of a single plane . In these more expansive contexts, closed polygonal chains that exist within higher-dimensional Euclidean space are still considered a type of polygon, specifically termed a skew polygon , even when all their vertices do not comfortably reside on a common plane . This broader, more inclusive perspective acknowledges the intrinsic structural pattern of the form, irrespective of its specific spatial embedding.

Ultimately, a polygon can be most accurately understood as a 2-dimensional instance, a specific manifestation, of the more encompassing and abstract concept of a polytope . The polytope generalizes this fundamental idea of bounded figures to any number of dimensions, from the familiar 2D polygon to 3D polyhedra and beyond. Just as a polygon meticulously defines a boundary and an interior in two dimensions, a polytope performs an analogous function in n-dimensional space. It’s almost as if geometry itself is built upon a recursive, self-similar structure, which, for all its inevitability, remains rather elegant. And, as is invariably the case with fundamental mathematical concepts, numerous other specialized generalizations of polygons have been meticulously defined and explored over time, each serving specific, often highly specialized, purposes.

Etymology

The very word “polygon” carries its meaning quite openly, a rather telling indicator of the straightforward observations that underpinned early geometric thought. Its linguistic lineage traces back to the Greek adjective πολύς (pronounced polús), which conveys the sense of ‘much’ or ‘many’, coupled with γωνία (pronounced gōnía), a term translating to ‘corner’ or ‘angle’. Thus, a polygon is, with rather literal precision, a “many-cornered” or “many-angled” figure. One might almost call it an obvious designation, devoid of any genuine linguistic intrigue. Intriguingly, some etymological scholars have also put forth the proposition that γόνυ (pronounced gónu), meaning ‘knee’, could potentially be the origin for the ‘gon’ component. This suggestion vaguely alludes to the characteristic bending or articulation that occurs at a polygon’s vertices . While a quaint and rather humanistic thought, it ultimately alters little of the fundamental, self-evident truth embedded within the term.

Classification

Polygons, much like any other structured entity in a universe that ostensibly values order, lend themselves readily to classification. Their myriad forms can be meticulously dissected, their inherent properties cataloged and assigned to distinct, logically labeled categories. This systematic classification transcends the mere act of counting sides, enabling a far more rigorous and nuanced comprehension of their diverse geometric manifestations.

Some different types of polygon

Number of sides

The most immediate, and frankly, least imaginative, method of classifying polygons is, quite predictably, by the sheer quantity of their sides. An n-gon is simply a polygon endowed with n sides—a system that, while undeniably efficient in its descriptive power, certainly makes no pretensions to poetic flair. This direct numerical count inherently dictates the corresponding number of vertices and, in the vast majority of cases, fundamentally influences the core geometric characteristics and potential behaviors of the figure.

Convexity and intersection

Beyond the elementary tally of sides, polygons are frequently characterized by their overarching morphology, specifically concerning their convexity or the precise nature of their non-convexity. This crucial distinction defines how the figure interacts with straight lines, whether it “bends” inward or maintains an outward-facing posture.

• Convex : A polygon earns the designation of convex if, and only if, any straight line drawn completely through its interior (and not merely grazing an edge or touching a vertex ) will, without exception, intersect its boundary at exactly two points. This singular property leads to a cascade of predictable characteristics: every single one of its interior angles must be strictly less than 180°. More intuitively, any line segment connecting two arbitrary points on the polygon’s boundary will lie entirely within, or precisely on, the boundary of the polygon itself. There are no inward indentations, no unexpected “caves” or recesses within a convex polygon . This condition is remarkably robust, holding true for polygons irrespective of the specific underlying geometry, extending beyond just Euclidean geometry . It represents a geometrically stable and inherently “well-behaved” form, the idealized, unblemished standard against which other polygons are often implicitly measured.

• Non-convex: As the name rather plainly, and perhaps a little uncreatively, implies, a non-convex polygon is simply any polygon that fails to satisfy the stringent criteria for convexity . This implies that at least one line can be found that intersects its boundary more than twice, or, equivalently, a line segment can be constructed between two points on its boundary that, at some intermediate point, passes outside the polygon’s explicitly defined interior. These are the polygons that exhibit geometric eccentricities, deviations from the perfectly rounded, outward-facing ideal.

• Simple : A simple polygon is definitively characterized by a boundary that does not intersect itself at any point other than its constituent vertices . It forms a single, continuous, unbroken loop. All convex polygons are, by logical extension and definition, simple. This category encompasses the most commonly visualized and extensively studied polygons, the standard forms one encounters in introductory geometry.

• Concave : This is a specific and important subset within the broader category of non-convex polygons, uniquely distinguished by being both non-convex and simple . The unmistakable hallmark of a concave polygon is the presence of at least one interior angle that measures greater than 180°. These are the polygons that visually appear to have been “pushed in” or “dented” from the exterior, creating an inward-facing angle.

• Star-shaped : A polygon is designated as star-shaped if its entire interior expanse is visible from at least one internal point, meaning that no part of the polygon’s edges obstructs the line of sight from this vantage point. This property necessarily implies that the polygon must be simple , though it can comfortably be either convex or concave . It follows that all convex polygons are, by their very nature, star-shaped , as every point within a convex polygon inherently possesses an unobstructed view of its entire interior.

• Self-intersecting : These polygons venture into more topologically complex and often visually confusing territory, where the boundary of the polygon itself crosses over itself. The term “complex” is occasionally, and rather imprecisely, employed as a direct contrast to “simple” in this context. However, such usage carries the risk of creating semantic confusion with the far more specialized and abstract concept of a complex polygon , which exists in the realm of the complex Hilbert plane and involves two complex dimensions—a notion significantly removed from mere visual intersection. Such nuanced distinctions, of course, are a constant source of amusement.

• Star polygon : A star polygon represents a specific, highly structured type of self-intersecting polygon that consistently exhibits a regular, often striking, pattern of self-intersection. It is a frequent point of confusion, or perhaps merely wishful thinking, to assume that a polygon can simultaneously be both a “star” (a star polygon ) and “star-shaped .” These are fundamentally distinct geometric properties: a star polygon is defined by its self-intersections, whereas a star-shaped polygon is always simple (non-self-intersecting) but may possess concave angles.

Equality and symmetry

Further, more sophisticated refinements in classification arise from examining the intrinsic symmetries and equalities within a polygon’s structure, properties that elevate certain forms beyond the realm of the geometrically mundane.

• Equiangular : Within an equiangular polygon , every single corner angle possesses the exact same angular measure. A testament to internal consistency, if nothing else.

• Equilateral : An equilateral polygon is defined by the property that all of its edges (or sides) are of precisely identical length. A pleasing balance of linear dimensions.

• Regular : The veritable paragon of polygonal forms, a regular polygon is one that achieves the ideal state of being simultaneously both equilateral and equiangular . These are the perfectly balanced, aesthetically harmonious shapes that intuitively spring to mind when one first contemplates the concept of a polygon.

• Cyclic : A cyclic polygon is a polygon whose vertices all lie with absolute precision upon the circumference of a single circle . This encompassing circle is conventionally referred to as the circumcircle of the polygon.

• Tangential : Conversely, a tangential polygon is distinguished by the property that all of its sides are precisely tangent to an inscribed circle that nestles perfectly within its interior.

• Isogonal or vertex-transitive : In an isogonal polygon, all of its corners are geometrically indistinguishable under the polygon’s symmetry operations ; they all reside within the same symmetry orbit . Such a polygon is necessarily both cyclic and equiangular , a consequence of its high degree of vertex symmetry.

• Isotoxal or edge-transitive : An isotoxal polygon is defined by the property that all of its sides are congruent and lie within the same symmetry orbit . This type of polygon is consequently both equilateral and tangential , a reflection of its edge symmetry.

The esteemed property of regularity, that highly sought-after state of geometric perfection, can also be elegantly defined through these symmetry attributes: a polygon attains regularity if and only if it is simultaneously both isogonal and isotoxal. Equivalently, it is regular if it is both cyclic and equilateral . A particularly intriguing departure from the typical visual order is the non-convex regular polygon, famously known as a regular star polygon , which manages to maintain its profound internal symmetries despite its captivating self-intersections.

Miscellaneous

Beyond these primary categorizations, additional properties allow for further, often more specialized, descriptions of polygons, catering to specific analytical needs.

• Rectilinear : A rectilinear polygon is defined by the rather rigid constraint that all of its sides meet exclusively at precise right angles. This means that every single one of its interior angles will measure either exactly 90° or exactly 270°. These are the polygons that adhere strictly to an orthogonal grid, devoid of any sloping or diagonal lines.

• Monotone with respect to a given line L: This classification delves into the topological behavior of the polygon. A polygon is considered monotone with respect to a given line L if every line orthogonal (perpendicular) to L intersects the polygon’s boundary no more than twice. This essentially implies that the polygon “flows” in a consistent direction relative to the line L, without any complex self-overlapping or excessive doubling back on itself, a useful property in computational geometry.

Properties and formulas

For the sake of unequivocal clarity and, frankly, a distinct aversion to unnecessary complications, the principles of Euclidean geometry are universally assumed throughout the subsequent discussion. Any deviations from this standard, should they arise, will be explicitly noted.

Partitioning an n -gon into n − 2 triangles

Angles

Any polygon, by its very inherent definition, possesses an equal number of corners and sides. Each of these corners is associated with several angular measures, the two most critically important and frequently discussed being the interior and exterior angles.

• Interior angle – The cumulative sum of the interior angles of any simple n-gon (a polygon possessing n sides) is invariably given by the formula ( n − 2) × π radians , which, when translated to the more commonly understood system, equates to ( n − 2) × 180 degrees . This fundamental geometric property arises from a rather elegant and oft-demonstrated observation: any simple n-gon can be meticulously decomposed, or triangulated, into precisely (n − 2) non-overlapping triangles . Given that each individual triangle itself possesses an angle sum of π radians or 180 degrees , the logic is undeniably sound. For a convex regular n-gon , where the ideal symmetry ensures all angles are identical, the measure of any single interior angle can be calculated with precision as:

  $$ \left(1-{\tfrac {2}{n}}\right)\pi $$ radians or $$ 180-{\tfrac {360}{n}} $$ degrees .

  The interior angles of regular star polygons —those geometrically fascinating, often mesmerizing, self-intersecting forms—were first subjected to systematic study by the brilliant Louis Poinsot . In the very paper where he meticulously described the four regular star polyhedra , he provided the definitive formula for these angles. For a regular $$ {\tfrac {p}{q}} $$ -gon (which is a p-gon characterized by a central density of q), each interior angle is precisely given by: $$ {\tfrac {\pi (p-2q)}{p}} $$ radians or $$ {\tfrac {180(p-2q)}{p}} $$ degrees . It’s a formula that, for all its abstractness, elegantly reveals the underlying order even within geometrically complex, self-intersecting structures.

• Exterior angle – The exterior angle at any given vertex of a polygon is, by its very definition, the supplementary angle to the interior angle at that same vertex . Consider, if you must, the mundane act of tracing the perimeter of a convex n-gon : as one navigates each successive corner , a specific angular amount is “turned.” This amount is precisely the exterior angle . If one completes a full circuit, traversing the entire perimeter of the polygon and returning to the starting point and orientation, one will have executed precisely one full turn . Consequently, the cumulative sum of all the exterior angles of any convex polygon must, without exception, always equal 360°. This elegant and intuitive principle can be rigorously generalized to encompass concave simple polygons , provided that exterior angles corresponding to turns in the opposite angular direction are correctly subtracted from the cumulative total. When tracing around an n-gon in its most general form, a definition that includes even the intricate self-intersecting polygons , the sum of the exterior angles (representing the total amount of rotation accumulated at the vertices ) can be any integer multiple d of 360°. In this context, d is a significant topological invariant known as the density or turning number of the polygon. For example, a pentagram exhibits a turning number of 2, resulting in a total exterior angle sum of 720°, whereas a degenerate angular “eight” or antiparallelogram can possess a turning number of 0, implying no net rotation.

Area

Coordinates of a non-convex pentagon

Calculating the area enclosed by a polygon is a fundamental geometric problem, one that becomes particularly intriguing and analytically demanding when confronting irregular or topologically complex shapes. In this section, for the sake of standardizing our approach, the vertices of the polygon under consideration are assumed to be ordered sequentially as $$ (x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1}) $$ . For mathematical convenience in certain formulas, it is a customary and practical convention to denote $$ (x_{n},y_{n}) = (x_{0},y_{0}) $$ , thereby gracefully closing the loop of coordinates.

Simple polygons

• Further information: Shoelace formula

If the polygon in question is non-self-intersecting—that is to say, it is a simple polygon —its signed area A can be computed with remarkable elegance and efficiency using the following formula, often attributed to the “shoelace” method:

$$ A={\frac {1}{2}}\sum {i=0}^{n-1}(x{i}y_{i+1}-x_{i+1}y_{i})\quad {\text{where }}x_{n}=x_{0}{\text{ and }}y_{n}=y_{0}, $$

Alternatively, for those who appreciate a touch more linear algebra in their geometry, one can employ determinants in a slightly more complex, yet equally valid, formulation:

$$ 16A^{2}=\sum {i=0}^{n-1}\sum {j=0}^{n-1}{\begin{vmatrix}Q{i,j}&Q{i,j+1}\Q_{i+1,j}&Q_{i+1,j+1}\end{vmatrix}}, $$

where $$ Q_{i,j} $$ represents the squared Euclidean distance between vertex $$ (x_{i},y_{i}) $$ and vertex $$ (x_{j},y_{j}) $$ . This specific formulation, while less frequently encountered in basic applications, hints at deeper mathematical connections within quadratic forms.

The resultant signed area is intrinsically dependent upon two factors: the specific ordering sequence of the vertices and the chosen orientation of the plane itself. Conventionally, the positive orientation is universally established by a counterclockwise rotation that maps the positive x-axis to the positive y-axis. If the vertices are listed in a counterclockwise sequence (i.e., in accordance with this positive orientation ), the signed area will, by definition, be positive. Conversely, should the vertices be ordered clockwise, the signed area will yield a negative value. In either scenario, the absolute value of the computed result provides the true, unambiguous geometric area of the polygon. This remarkably versatile formula is widely known as the shoelace formula or the surveyor’s formula, a moniker derived from the characteristic crisscrossing pattern of its terms, which bears a visual resemblance to the intricate lacing of a shoe. It is a deceptively simple yet incredibly practical piece of mathematics, surprisingly useful for something so conceptually abstract.

The area A of a simple polygon can also be determined if one possesses comprehensive knowledge of the lengths of its sides, denoted as a₁, a₂, …, a_n, in conjunction with its corresponding exterior angles , θ₁, θ₂, …, θ_n. The formula, while considerably more elaborate in its structure, offers an alternative analytical perspective:

$$ {\begin{aligned}A={\frac {1}{2}}(a_{1}[a_{2}\sin(\theta {1})+a{3}\sin(\theta _{1}+\theta {2})+\cdots +a{n-1}\sin(\theta _{1}+\theta {2}+\cdots +\theta {n-2})]\{}+a{2}[a{3}\sin(\theta {2})+a{4}\sin(\theta _{2}+\theta {3})+\cdots +a{n-1}\sin(\theta {2}+\cdots +\theta {n-2})]\{}+\cdots +a{n-2}[a{n-1}\sin(\theta _{n-2})]).\end{aligned}}} $$

This particular formula was meticulously described by A.M. Lopshits in 1963, thus adding another specialized, albeit less intuitive, tool to the geometrician’s ever-expanding arsenal.

For polygons that can be precisely drawn on an equally spaced grid, where all of their vertices perfectly align with the grid points, a delightfully simple and remarkably elegant formula emerges: Pick’s theorem . This theorem states that the polygon’s area can be calculated directly from the number of interior grid points (I) and boundary grid points (B): A = I + B/2 − 1. It offers a charmingly direct and intuitive method to quantify space for those who appreciate the inherent order of a well-defined lattice.

Furthermore, for every polygon, irrespective of its complexity, characterized by a perimeter p and an area A, the fundamental isoperimetric inequality holds true:

$$ p^{2}>4\pi A $$

This profound inequality unveils a deep and immutable relationship, rigorously demonstrating that among all possible shapes that share a given perimeter, the circle is the unique figure that encloses the absolute maximum area . All other polygons, by their very nature, are inherently less efficient in this regard, a geometric truth that underscores the circle’s optimal packing efficiency.

In a more abstract but equally fascinating vein, the Bolyai–Gerwien theorem asserts a remarkable equivalence: for any two simple polygons that possess precisely the same area , the first polygon can invariably be meticulously dissected into a finite number of smaller polygonal pieces. These pieces, once separated, can then be reassembled, without any overlap or gaps, to perfectly form the second polygon. This theorem speaks to a deep, transformative equivalence between shapes that, despite differing forms, ultimately share an identical measure of space, a testament to the powerful implications of geometric dissection.

It is a pervasive misconception, frequently leading to erroneous assumptions, that the lengths of a polygon’s sides are, by themselves, sufficient to uniquely determine its area . In general, this is demonstrably not the case; a polygon can often be “flexed” or reshaped into different configurations while maintaining the exact same side lengths, thereby altering its enclosed area . This is why you can’t just measure the perimeter and expect to know everything. However, if a polygon is both simple and cyclic —meaning its vertices all lie perfectly on the circumference of a single circle —then, and only then, do its side lengths uniquely and unambiguously define its area . Furthermore, among all n-gons constructed with a specified set of side lengths, the one that encloses the largest possible area is invariably the cyclic one. And, pushing this principle further, of all n-gons sharing a given perimeter, the polygon that achieves the maximum area is none other than the regular polygon (which, by its very definition, is also cyclic ). It seems regularity, once again, reliably leads to optimal efficiency and maximal spatial enclosure.

Regular polygons

For the mathematically inclined, numerous specialized and often considerably more concise formulas apply specifically to the calculation of areas for regular polygons , significantly simplifying computations for these geometrically ideal forms.

The area A of a regular polygon can be expressed with elegant simplicity in terms of r, the radius of its inscribed circle (which is the largest circle that can fit entirely within the polygon while being tangent to all of its sides), and p, its total perimeter:

$$ A={\tfrac {1}{2}}\cdot p\cdot r. $$

This specific radius r is also frequently referred to as the polygon’s apothem and is often, for brevity, represented by the variable a.

Alternatively, the area A of a regular n-gon can be calculated with equal precision using R, the radius of its circumscribed circle (the unique circle that passes through all of the vertices of the regular n-gon ):

$$ A=R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=R^{2}\cdot n\cdot \sin {\frac {\pi }{n}}\cdot \cos {\frac {\pi }{n}} $$

These formulas vividly underscore the profound and intrinsic connection between regular polygons and circles . Indeed, as the number of sides n of a regular polygon approaches infinity (a theoretical limit, of course), the polygon itself asymptotically approaches the perfect form of a circle . Consequently, its area formula converges precisely to the familiar area of a circle with radius R: $$ \lim _{n\to +\infty }R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=\pi \cdot R^{2} $$ . A rather neat demonstration of the power of calculus and limits, if you happen to find such convergence satisfying.

Self-intersecting

The concept of area becomes significantly more ambiguous, and indeed requires meticulous definition, when one ventures into the territory of self-intersecting polygons . The very act of the polygon’s boundary crossing itself fundamentally complicates what constitutes “inside” and “outside,” leading to at least two distinct and often conflicting interpretations, each yielding a different quantitative result.

• One prevalent definition employs the same fundamental formulas utilized for simple polygons , but with a critical modification: particular regions within the polygon are assigned a “density” factor, which effectively multiplies their contribution to the overall total area . For example, the central convex pentagon that forms the core of a pentagram is considered to have a density of 2, meaning its area is counted twice in the summation. In the case of a cross-quadrilateral (a figure resembling a figure “8”), the two triangular regions formed by the self-intersection are assigned opposite-signed densities. When their areas are algebraically summed, this can, counter-intuitively, result in a total area of zero for the entire figure, a result that is mathematically consistent within this framework but often bewildering to the uninitiated.

• A second, more pragmatic approach treats the enclosed regions solely as point sets. Under this definition, the area is simply the measure of the total unique expanse of the plane that is covered by the polygon, irrespective of how many times the boundary traverses or crosses over itself. This method effectively corresponds to calculating the area of one or more simple polygons that share the exact same outer outline as the self-intersecting polygon , thereby deliberately ignoring the internal self-intersections for the sole purpose of defining the encompassing boundary. In the specific instance of the cross-quadrilateral, this approach would treat it as two distinct simple triangles , and their absolute areas would then be summed to yield the total covered region. This method often aligns more closely with common, intuitive notions of “space occupied” or “paint coverage.”

Centroid

The centroid of a polygon represents its geometric center, its precise point of balance. As with the calculation of area , its determination depends critically on whether one is considering the solid, enclosed polygon or merely the discrete set of its vertices . Using the same established convention for vertex coordinates as defined in the preceding section, the coordinates of the centroid $$ (C_x, C_y) $$ of a solid simple polygon are given by the following formulas:

$$ C_{x}={\frac {1}{6A}}\sum {i=0}^{n-1}(x{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}), $$

$$ C_{y}={\frac {1}{6A}}\sum {i=0}^{n-1}(y{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}). $$

It is absolutely crucial to note that in these formulas, the signed value of the area $$ A $$ (as correctly obtained from the shoelace formula ) must be utilized. Failure to adhere to this detail will, with predictable certainty, yield incorrect and misleading results.

For triangles (n = 3), a unique and rather convenient simplification occurs: the centroid of the vertices and the centroid of the solid triangular shape coincide perfectly, occupying the identical point in space. However, this convenient equivalence is generally not true for polygons with n > 3 sides. For polygons possessing more than three sides, the two centroids are, in fact, distinct entities. The centroid of the vertex set of a polygon with n vertices is calculated simply as the arithmetic average of their respective coordinates:

$$ c_{x}={\frac {1}{n}}\sum {i=0}^{n-1}x{i}, $$

$$ c_{y}={\frac {1}{n}}\sum {i=0}^{n-1}y{i}. $$

This subtle but significant distinction between the geometric center of the boundary points and the geometric center of the enclosed mass is an important nuance, often overlooked by those who prefer their mathematical truths to be uncomplicated and universally applicable. Alas, reality is rarely so accommodating.

Generalizations

The core idea of a polygon, while seemingly straightforward in its familiar Euclidean plane manifestation, has been ingeniously expanded and generalized in a multitude of ways by mathematicians, relentlessly pushing the conceptual boundaries of what a “many-sided figure” can truly encompass. Because, apparently, the simple, observable reality was deemed insufficient.

• A spherical polygon dares to venture beyond the confines of the flat plane , existing instead upon the curved, expansive surface of a sphere . It is precisely defined as a circuit of arcs of great circles (which serve as its “sides”) connecting a sequence of vertices that are all situated on the sphere’s surface . This particular generalization is especially noteworthy because it permits the very existence of the digon , a polygon possessing only two sides and two corners—a configuration utterly, geometrically impossible in a flat Euclidean plane . Spherical polygons are far from mere abstract curiosities; they play a profoundly critical role in practical fields such as cartography (the intricate art and science of map making), where the Earth’s inherently curved surface necessitates such geometric adaptations, and they are also fundamental in Wythoff’s construction of the uniform polyhedra .

• A skew polygon is a polygon that resolutely refuses to be contained within a single flat plane . Instead, its vertices and edges elegantly zigzag through three (or even more) spatial dimensions. While its edges remain straight line segments and it still forms a closed polygonal chain , the entire figure does not lie coplanar. The Petrie polygons of the regular polytopes (which are higher-dimensional analogues of polygons and polyhedra ) serve as particularly well-known and visually striking examples of skew polygons .

• An apeirogon , a term derived from the Greek “apeiros” meaning infinite, is conceptually a degenerate polygon of infinitely many sides. It represents an infinite sequence of sides and angles which, by its very nature, cannot be “closed” in the traditional sense, yet simultaneously possesses no definitive ends, extending indefinitely in both directions. It is, in essence, what happens when the concept of “many” is pushed to its absolute, unbounded limit.

• A skew apeirogon takes the already boundless concept of the apeirogon and elevates it out of the confining plane . It is an infinite sequence of sides and angles that, much like its finite skew polygon counterpart, does not lie within a single flat plane , traversing infinite space.

• A polygon with holes describes a region of the plane that, while geometrically connected (meaning one can travel between any two points within it without leaving the region), is topologically “multiply-connected.” It possesses one external boundary, which defines its overall encompassing shape, and one or more interior boundaries, which constitute the “holes” within its structure. Conceptually, one might imagine a flat doughnut or a washer, but with straight, polygonal sides.

• A complex polygon represents a more abstract and algebraic generalization. It is a configuration that maintains an analogy to an ordinary polygon but exists within the highly abstract realm of the complex plane —a mathematical space fundamentally composed of two real dimensions and two imaginary dimensions. This particular generalization catapults polygons far beyond simple visual representation and firmly into the domain of abstract algebra and higher-dimensional geometry, a place where visualization is often less helpful than rigorous definition.

• An abstract polygon fundamentally strips away all geometric properties, focusing instead purely on the combinatorial relationships and connectivity between its constituent elements. It is formally defined as an algebraic partially ordered set that meticulously represents the various elements of a polygon (such as its sides and vertices ) and their intricate interconnections. A “real” geometric polygon, the kind one can sketch or readily visualize, is then considered a “realization” or a specific geometric manifestation of its associated abstract polygon . Depending on the specific mapping or “realization” employed, all the geometric generalizations described herein can be understood as tangible manifestations of their underlying abstract polygon structures.

• A polyhedron extends the fundamental concept of a polygon into the third spatial dimension. It is a three-dimensional solid that is bounded by flat polygonal faces, thus serving as the direct three-dimensional analogue to a polygon in two dimensions. These bounding faces are, by definition, themselves polygons. The corresponding geometric shapes in four or higher dimensions are generally referred to as polytopes . (It is worth noting, for the sake of meticulous accuracy, that in some alternative conventions within the field, the terms “polyhedron ” and “polytope ” are used more broadly to refer to such figures in any dimension, with the primary distinguishing characteristic often being that a polytope is necessarily bounded, whereas a polyhedron might, under certain definitions, extend infinitely.)

Naming

The convention for naming polygons, much like their foundational definitions, is deeply rooted in Late Latin and Greek linguistic traditions. The word polygōnum (a noun in Late Latin ) originates from the Greek πολύγωνον (polygōnon/polugōnon), which, when employed as a noun, is the neuter form of the masculine adjective πολύγωνος (polygōnos/polugōnos), meaning “many-angled.” A rather literal and uninspired lineage, if you ask me.

Individual polygons are typically named, and frequently classified, based on the precise number of sides they possess. This systematic approach involves combining a Greek -derived numerical prefix with the ubiquitous suffix “-gon.” For instance, a 5-sided polygon is universally known as a pentagon , and a 12-sided polygon is a dodecagon . However, there are a few notable and rather persistent exceptions to this otherwise consistent system: the triangle (3 sides), quadrilateral (4 sides), and nonagon (9 sides) employ Latin or unique English roots for their common names, though their Greek-derived alternatives (trigon, tetragon, and enneagon, respectively) are also formally recognized, if considerably less frequently encountered in general usage.

Beyond the relatively common decagons (10-sided) and dodecagons (12-sided), mathematicians, perhaps reaching the limits of their linguistic creativity or simply prioritizing clarity, generally resort to a simpler, more pragmatic numerical notation. Thus, one typically refers to a 17-gon or a 257-gon, a clear indication that directness often, and quite rightly, trumps convoluted linguistic gymnastics in the pursuit of mathematical precision.

Nevertheless, exceptions to this numerical shorthand persist, particularly for side counts that are readily expressible in a more verbal form (e.g., “20-gon” for icosagon or “30-gon” for triacontagon ), especially when these terms are used by non-mathematicians who might find the formal Greek-derived names less immediately intuitive. Furthermore, some specialized polygons have, over time, earned their own distinct and culturally resonant names. A prime example is the regular star pentagon , which is far more widely recognized as a pentagram or pentacle, a name imbued with significant cultural and historical weight far beyond its purely geometric definition.

Polygon names and miscellaneous properties

| Name ## The Enduring Geometry: An Exploration of Polygons

For other uses, see Polygon (disambiguation) .

Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.

In the meticulously ordered, and frankly, often quite demanding, universe of geometry , the polygon stands as an absolutely fundamental concept: a distinct plane figure meticulously constructed from a finite, sequential arrangement of straight line segments . These segments are not merely placed haphazardly; they are rigorously connected end-to-end, forming what is precisely termed a closed polygonal chain . This characteristic of closure is not merely a descriptive detail; it is an utterly critical and immutable requirement, serving to unequivocally distinguish a polygon—a truly enclosed form—from a simple, open sequence of lines that merely meanders without resolution. To genuinely earn the designation of a polygon, the figure must, without ambiguity, completely enclose a specific region, its geometric path returning with exacting precision to its initial point of origin.

The individual, rectilinear line segments that collectively compose this closed polygonal chain are formally recognized as its edges , or, in more casual parlance, its sides. The exact points in space where any two consecutive edges converge and meet are precisely termed the polygon’s vertices or corners. The system of nomenclature employed for polygons is, mercifully, often quite direct and intuitive, primarily dictated by the sheer quantity of these sides: an n-gon denotes a polygon possessing exactly n sides. For illustrative purposes, consider a triangle , which one might regard as the irreducible, simplest Euclidean polygon; it is, by definition, a 3-gon. Following this logical and consistent progression, a quadrilateral is a 4-gon, a pentagon a 5-gon, and so forth, a system that, while admittedly lacking in any particular poetic grandeur, is undeniably clear, unambiguous, and remarkably efficient for classification.

Among the myriad manifestations of polygons, the simple polygon occupies a position of particular prominence, largely due to its predictable and exceptionally well-behaved nature. This specific classification is assigned to any polygon whose boundary, in its intricate construction, does not intersect itself at any point other than the shared endpoints of consecutive segments within the polygonal chain. All convex polygons , for instance, are, by their very definition, simple. When the term “simple polygon ” is invoked, it most commonly refers to the clear demarcation, the boundary, that elegantly encloses a single, distinct, and unambiguous region of the plane . This enclosed region itself is often referred to as a solid polygon, or its internal expanse, known as its body, or more explicitly, a polygonal region or polygonal area. In practical, everyday contexts, particularly when the discussion implicitly focuses on these uncomplicated, non-self-intersecting forms, the generic term “polygon” frequently serves as a concise shorthand, referring interchangeably to either the simple polygon (the boundary itself) or the solid polygon (the filled interior). This subtle but critical distinction between the bounding line and the space it defines, while often regrettably glossed over in casual conversation, is fundamentally important for precise geometric understanding and avoids a great deal of unnecessary confusion.

However, the geometric landscape of polygons is not solely populated by such pristine, non-intersecting forms. A polygonal chain possesses the inherent, almost rebellious, geometric freedom to cross over itself, thereby giving rise to more intricate and topologically complex figures such as star polygons and a broader array of self-intersecting polygons . These fascinating figures, while faithfully adhering to the core definition of connected line segments and ultimate closure, introduce significant complexities in how their “interior” is conceptually defined and quantitatively measured; indeed, the very notion of an interior becomes debatable. Furthermore, certain advanced mathematical frameworks extend the very definition of a polygon beyond the strict confines of a single plane . In these more expansive and abstract contexts, closed polygonal chains that exist within higher-dimensional Euclidean space are still considered a type of polygon, specifically termed a skew polygon , even when all their vertices do not comfortably reside on a common plane . This broader, more inclusive perspective acknowledges the intrinsic structural pattern of the form, irrespective of its specific spatial embedding.

Ultimately, a polygon can be most accurately understood as a 2-dimensional instance, a specific manifestation, of the more encompassing and abstract concept of a polytope . The polytope generalizes this fundamental idea of bounded figures to any number of dimensions, from the familiar 2D polygon to 3D polyhedra and beyond. Just as a polygon meticulously defines a boundary and an interior in two dimensions, a polytope performs an analogous function in n-dimensional space. It’s almost as if geometry itself is built upon a recursive, self-similar structure, which, for all its inevitability, remains rather elegant. And, as is invariably the case with fundamental mathematical concepts, numerous other specialized generalizations of polygons have been meticulously defined and explored over time, each serving specific, often highly specialized, purposes—because, apparently, the simple reality of flat, closed shapes wasn’t quite enough.

Etymology

The very word “polygon” carries its meaning quite openly, a rather telling indicator of the straightforward, almost painfully obvious, observations that underpinned early geometric thought. Its linguistic lineage traces back to the Greek adjective πολύς (pronounced polús), which conveys the sense of ‘much’ or ‘many’, coupled with γωνία (pronounced gōnía), a term translating to ‘corner’ or ‘angle’. Thus, a polygon is, with rather literal precision, a “many-cornered” or “many-angled” figure. One might almost call it an obvious designation, devoid of any genuine linguistic intrigue or philosophical depth. Intriguingly, some etymological scholars have also put forth the proposition that γόνυ (pronounced gónu), meaning ‘knee’, could potentially be the origin for the ‘gon’ component. This suggestion vaguely alludes to the characteristic bending or articulation that occurs at a polygon’s vertices . While a quaint and rather humanistic thought, it ultimately alters little of the fundamental, self-evident truth embedded within the term.

Classification

Polygons, much like any other structured entity in a universe that ostensibly values order, lend themselves readily to classification. Their myriad forms can be meticulously dissected, their inherent properties cataloged and assigned to distinct, logically labeled categories. This systematic classification transcends the elementary act of merely counting sides, enabling a far more rigorous and nuanced comprehension of their diverse geometric manifestations. It’s a way to impose order, even if the polygons themselves couldn’t care less.

Some different types of polygon

Number of sides

The most immediate, and frankly, least imaginative, method of classifying polygons is, quite predictably, by the sheer quantity of their sides. An n-gon is simply a polygon endowed with n sides—a system that, while undeniably efficient in its descriptive power, certainly makes no pretensions to poetic flair. This direct numerical count inherently dictates the corresponding number of vertices and, in the vast majority of cases, fundamentally influences the core geometric characteristics and potential behaviors of the figure. This is where most people stop, assuming they’ve grasped the entirety of the concept. They haven’t.

Convexity and intersection

Beyond the elementary tally of sides, polygons are frequently characterized by their overarching morphology, specifically concerning their convexity or the precise nature of their non-convexity. This crucial distinction defines how the figure interacts with straight lines, whether it “bends” inward or maintains an outward-facing posture, revealing its inner geometric disposition.

• Convex : A polygon earns the designation of convex if, and only if, any straight line drawn completely through its interior (and not merely grazing an edge or touching a vertex ) will, without exception, intersect its boundary at exactly two points. This singular, defining property leads to a cascade of predictable characteristics: every single one of its interior angles must be strictly less than 180°. More intuitively, and perhaps more accessibly, any line segment connecting two arbitrary points on the polygon’s boundary will lie entirely within, or precisely on, the boundary of the polygon itself. There are no inward indentations, no unexpected “caves” or recesses within a convex polygon ; it presents a uniformly “bulging” or outward-facing form. This condition is remarkably robust, holding true for polygons irrespective of the specific underlying geometry, extending beyond just Euclidean geometry . It represents a geometrically stable and inherently “well-behaved” form, the idealized, unblemished standard against which other polygons are often implicitly measured. It’s the polygon that doesn’t try to surprise you.

• Non-convex: As the name rather plainly, and perhaps a little uncreatively, implies, a non-convex polygon is simply any polygon that fails to satisfy the stringent criteria for convexity . This implies that at least one line can be found that intersects its boundary more than twice, or, equivalently, a line segment can be constructed between two points on its boundary that, at some intermediate point, passes outside the polygon’s explicitly defined interior. These are the polygons that exhibit geometric eccentricities, deviations from the perfectly rounded, outward-facing ideal. They are, in a word, imperfect.

• Simple : A simple polygon is definitively characterized by a boundary that does not intersect itself at any point other than its constituent vertices . It forms a single, continuous, unbroken loop, enclosing a clear, unambiguous region. All convex polygons are, by logical extension and definition, simple. This category encompasses the most commonly visualized and extensively studied polygons, the standard forms one encounters in introductory geometry. These are the polygons that don’t complicate things unnecessarily.

• Concave : This is a specific and important subset within the broader category of non-convex polygons, uniquely distinguished by being both non-convex and simple . The unmistakable hallmark of a concave polygon is the presence of at least one interior angle that measures greater than 180°. These are the polygons that visually appear to have been “pushed in” or “dented” from the exterior, creating an inward-facing angle that breaks the smooth convexity. They have a “bite” taken out of them, so to speak.

• Star-shaped : A polygon is designated as star-shaped if its entire interior expanse is visible from at least one internal point, meaning that no part of the polygon’s edges obstructs the line of sight from this vantage point. This property necessarily implies that the polygon must be simple , though it can comfortably be either convex or concave . It follows that all convex polygons are, by their very nature, star-shaped , as every point within a convex polygon inherently possesses an unobstructed view of its entire interior. This is not to be confused with a “star polygon,” which is a different, more complex beast.

• Self-intersecting : These polygons venture into more topologically complex and often visually confusing territory, where the boundary of the polygon itself crosses over itself. The term “complex” is occasionally, and rather imprecisely, employed as a direct contrast to “simple” in this context. However, such usage carries the distinct risk of creating semantic confusion with the far more specialized and abstract concept of a complex polygon , which exists in the realm of the complex Hilbert plane and involves two complex dimensions—a notion significantly removed from mere visual intersection. Such nuanced distinctions, of course, are a constant source of amusement for those who appreciate precision, and a constant headache for everyone else.

• Star polygon : A star polygon represents a specific, highly structured type of self-intersecting polygon that consistently exhibits a regular, often striking, pattern of self-intersection. It is a frequent point of confusion, or perhaps merely wishful thinking on the part of the geometrically uninitiated, to assume that a polygon can simultaneously be both a “star” (a star polygon ) and “star-shaped .” These are fundamentally distinct geometric properties: a star polygon is defined by its self-intersections, whereas a star-shaped polygon is always simple (non-self-intersecting) but may possess concave angles. One is a curiosity of intersection, the other a peculiarity of indentation.

Equality and symmetry

Further, more sophisticated refinements in classification arise from examining the intrinsic symmetries and equalities within a polygon’s structure, properties that elevate certain forms beyond the realm of the geometrically mundane. These are the polygons that exhibit a certain predictable elegance.

• Equiangular : Within an equiangular polygon , every single corner angle possesses the exact same angular measure. A testament to internal consistency, if nothing else. All its turns are identical.

• Equilateral : An equilateral polygon is defined by the property that all of its edges (or sides) are of precisely identical length. A pleasing balance of linear dimensions, a uniform stride around its perimeter.

• Regular : The veritable paragon of polygonal forms, a regular polygon is one that achieves the ideal state of being simultaneously both equilateral and equiangular . These are the perfectly balanced, aesthetically harmonious shapes that intuitively spring to mind when one first contemplates the concept of a polygon. They are, in essence, the “perfect” polygons, as far as Euclidean geometry is concerned.

• Cyclic : A cyclic polygon is a polygon whose vertices all lie with absolute precision upon the circumference of a single circle . This encompassing circle is conventionally referred to as the circumcircle of the polygon. The circle “goes around” it perfectly.

• Tangential : Conversely, a tangential polygon is distinguished by the property that all of its sides are precisely tangent to an inscribed circle that nestles perfectly within its interior. The circle “touches” all its sides.

• Isogonal or vertex-transitive : In an isogonal polygon, all of its corners are geometrically indistinguishable under the polygon’s symmetry operations ; they all reside within the same symmetry orbit . This means that any vertex can be mapped onto any other vertex by a symmetry operation of the polygon. Such a polygon is necessarily both cyclic and equiangular , a consequence of its high degree of vertex symmetry.

• Isotoxal or edge-transitive : An isotoxal polygon is defined by the property that all of its sides are congruent and lie within the same symmetry orbit . This means any edge can be mapped onto any other edge by a symmetry operation. This type of polygon is consequently both equilateral and tangential , a reflection of its edge symmetry.

The esteemed property of regularity, that highly sought-after state of geometric perfection, can also be elegantly defined through these symmetry attributes: a polygon attains regularity if and only if it is simultaneously both isogonal and isotoxal. Equivalently, it is regular if it is both cyclic and equilateral . A particularly intriguing departure from the typical visual order is the non-convex regular polygon, famously known as a regular star polygon , which manages to maintain its profound internal symmetries despite its captivating self-intersections. It’s a testament to the fact that even complex forms can exhibit perfect order.

Miscellaneous

Beyond these primary categorizations, additional properties allow for further, often more specialized, descriptions of polygons, catering to specific analytical needs or architectural styles.

• Rectilinear : A rectilinear polygon is defined by the rather rigid constraint that all of its sides meet exclusively at precise right angles. This means that every single one of its interior angles will measure either exactly 90° or exactly 270°. These are the polygons that adhere strictly to an orthogonal grid, devoid of any sloping or diagonal lines, often found in architectural blueprints or early computer graphics.

• Monotone with respect to a given line L: This classification delves into the topological behavior of the polygon. A polygon is considered monotone with respect to a given line L if every line orthogonal (perpendicular) to L intersects the polygon’s boundary no more than twice. This essentially implies that the polygon “flows” in a consistent direction relative to the line L, without any complex self-overlapping or excessive doubling back on itself, a useful property in computational geometry for simplifying algorithms like point-in-polygon tests.

Properties and formulas

For the sake of unequivocal clarity and, frankly, a distinct aversion to unnecessary complications, the principles of Euclidean geometry are universally assumed throughout the subsequent discussion. Any deviations from this standard, should they arise, will be explicitly noted. Don’t worry, it’s not going to get too complicated, for now.

Partitioning an n -gon into n − 2 triangles

Angles

Any polygon, by its very inherent definition, possesses an equal number of corners and sides. Each of these corners is associated with several angular measures, the two most critically important and frequently discussed being the interior and exterior angles. These angles dictate how a polygon fits together, or how it turns.

• Interior angle – The cumulative sum of the interior angles of any simple n-gon (a polygon possessing n sides) is invariably given by the formula ( n − 2) × π radians , which, when translated to the more commonly understood system, equates to ( n − 2) × 180 degrees . This fundamental geometric property arises from a rather elegant and oft-demonstrated observation: any simple n-gon can be meticulously decomposed, or triangulated, into precisely (n − 2) non-overlapping triangles . Given that each individual triangle itself possesses an angle sum of π radians or 180 degrees , the logic is undeniably sound. It’s a neat piece of deductive reasoning that simplifies the complex into the elementary. For a convex regular n-gon , where the ideal symmetry ensures all angles are identical, the measure of any single interior angle can be calculated with precision as:

  $$ \left(1-{\tfrac {2}{n}}\right)\pi $$ radians or $$ 180-{\tfrac {360}{n}} $$ degrees .

  The interior angles of regular star polygons —those geometrically fascinating, often mesmerizing, self-intersecting forms—were first subjected to systematic study by