Pentagon

Right. Let's get this over with. You want to know about shapes with five sides. Fascinating. Truly. Here’s the breakdown, precisely as it should be.

Shape with five sides

This article is dedicated to the geometric figure itself. For the imposing headquarters of the United States Department of Defense, you’ll need to consult a different source. And if you’re looking for a broader exploration of things named "pentagon," there’s a disambiguation page for that, too.

Pentagon

A cyclic polygon, meaning all its vertices lie on a single circle.

• Edges: 5
• Vertices: 5

In the grand, ancient language of Greek, the word "pentagon" is derived from "pente," meaning 'five,' and "gonia," meaning 'angle.' So, quite literally, it’s a five-angled thing. In the realm of geometry, a pentagon is any polygon that boasts five sides, or, as some might call it, a 5-gon. If we’re talking about a simple, unbroken pentagon, the sum of its internal angles will always add up to a neat 540°.

Now, understand that a pentagon isn't always a polite, well-behaved shape. It can be simple, or it can be the sort of shape that crosses itself, a self-intersecting polygon. When a regular pentagon decides to get fancy and intersect itself, it’s known as a pentagram, or a star pentagon. Think of it as the pentagon's wilder, more dramatic alter ego.

Regular pentagons

The epitome of pentagonal perfection, the regular pentagon.

• Type: Regular polygon
• Edges and vertices: 5
• Schläfli symbol: {5}
• Coxeter–Dynkin diagrams: (None specified, which is its own kind of statement)
• Symmetry group: Dihedral (D₅), order 10. It has five lines of reflectional symmetry and rotational symmetry of order 5, meaning it looks the same after rotations of 72°, 144°, 216°, and 288°.
• Internal angle: 108°
• Properties: Convex, cyclic, equilateral, isogonal, isotoxal. It’s a shape that ticks all the boxes for polite geometric behavior.
• Dual polygon: Self. It’s its own inverse. How introspective.

For a regular pentagon with a side length denoted as t ( $\displaystyle t$ ), the other key measurements are:

• Circumradius (R): The distance from the center to a vertex.
• Inscribed circle radius (r): Also known as the apothem, the distance from the center to the midpoint of a side.
• Height (H): The distance from one side to the opposite vertex.
• Width/Diagonal (W or D): The distance between the two farthest points, which is the length of a diagonal.

These are related by the following rather precise formulas:

$H = \frac{\sqrt{5+2\sqrt{5}}}{2} t \approx 1.539 t$ $W = D = \frac{1+\sqrt{5}}{2} t \approx 1.618 t$

The width, interestingly, is also related to the height:

$W = \sqrt{2 - \frac{2}{\sqrt{5}}} H \approx 1.051 H$

The circumradius, the distance to the vertices, is:

$R = \sqrt{\frac{5+\sqrt{5}}{10}} t \approx 0.8507 t$

And the diagonal (D), which is twice the circumradius times the cosine of 18 degrees, or approximately 1.902 times the circumradius:

$D = R \sqrt{\frac{5+\sqrt{5}}{2}} = 2R \cos 18^{\circ} = 2R \cos \frac{\pi}{10} \approx 1.902 R$

The area (A) of a regular pentagon with side length t is a bit more involved:

$A = \frac{t^2 \sqrt{25+10\sqrt{5}}}{4} = \frac{5t^2 \tan 54^{\circ}}{4} = \frac{\sqrt{5(5+2\sqrt{5})}}{4} t^2 = \frac{t^2 \sqrt{4\varphi^5+3}}{4} \approx 1.720 t^2$

Here, $\varphi$ represents the golden ratio, a recurring theme with pentagons. If you know the circumradius (R) instead, the side length (t) can be found:

$t = R \sqrt{\frac{5-\sqrt{5}}{2}} = 2R \sin 36^{\circ} = 2R \sin \frac{\pi}{5} \approx 1.176 R$

And the area in terms of R:

$A = \frac{5R^2}{4} \sqrt{\frac{5+\sqrt{5}}{2}}$

Compared to its circumscribed circle (area $\displaystyle \pi R^2$ ), the regular pentagon occupies about 75.68% of that space. Not bad for a five-sided figure.

Derivation of the area formula

The general formula for the area of any regular polygon is $A = \frac{1}{2} Pr$, where P is the perimeter and r is the inradius (or apothem). For our regular pentagon, with perimeter $P = 5t$ and apothem $r = \frac{t}{2 \tan(\pi/5)}$, we arrive at:

$A = \frac{1}{2} \cdot 5t \cdot \frac{t \tan \left( \frac{3\pi}{10} \right)}{2} = \frac{5t^2 \tan \left( \frac{3\pi}{10} \right)}{4}$

Inradius

The inradius, or apothem (r), of a regular pentagon is related to its side length t by:

$r = \frac{t}{2 \tan \left( \frac{\pi}{5} \right)} = \frac{t}{2\sqrt{5-\sqrt{20}}} \approx 0.6882 t$

Chords from the circumscribed circle to the vertices

For a regular pentagon inscribed in a circle, if you pick any point P on the circumcircle between two vertices, say B and C, the sum of the distances from P to the two vertices furthest away (PA and PD) will equal the sum of the distances to the other three vertices (PB + PC + PE). It’s a rather elegant geometric property.

Point in plane

Let's say you have a regular pentagon with circumradius R, and L is the distance from its center to some point in the plane. If $d_i$ are the distances from that point to each of the pentagon's five vertices, then there are some rather fascinating relationships:

$\sum_{i=1}^{5} d_i^2 = 5(R^2+L^2)$ $\sum_{i=1}^{5} d_i^4 = 5((R^2+L^2)^2+2R^2L^2)$ $\sum_{i=1}^{5} d_i^6 = 5((R^2+L^2)^3+6R^2L^2(R^2+L^2))$ $\sum_{i=1}^{5} d_i^8 = 5((R^2+L^2)^4+12R^2L^2(R^2+L^2)^2+6R^4L^4)$

And if $d_i$ are the distances from the vertices of a regular pentagon to a point on its circumcircle, then:

$3\left(\sum_{i=1}^{5} d_i^2\right)^2 = 10 \sum_{i=1}^{5} d_i^4$

It’s a bit much, I know. Just more proof that geometry isn't always straightforward.

Geometrical constructions

The fact that the regular pentagon can be constructed using only a compass and straightedge is significant, especially since 5 is a Fermat prime. There are several known methods.

Richmond's method

A method described by Richmond, and elaborated upon in Cromwell's book Polyhedra. Imagine a unit circle. Find the midpoint M of a radius. Connect M to the point D directly above the center. Bisect the angle CMD. This bisector hits the vertical axis at Q. A horizontal line through Q intersects the circle at P. The chord PD is the side of the inscribed pentagon. The derivation involves some Pythagoras' theorem and half-angle formulas, resulting in the side length being $\frac{\sqrt{5}-1}{4}$ times the diameter, or more precisely, $\frac{\sqrt{5}-1}{2}$ if the radius is 1. It confirms that $m\angle CDP = 54^{\circ}$, as expected for a regular pentagon inscribed in a circle.

Carlyle circles

This method, derived from finding roots of a quadratic equation, offers a visual approach. You start with a circle, mark points, find midpoints, and draw more circles. Specifically, you draw a circle centered at the midpoint M of a radius OB, passing through point A (on the circle, directly above the center). The intersection of this new circle with the line OB, inside the original circle, is point W. A circle centered at W with radius OA will intersect the original circle, giving you two vertices. The fifth vertex is simply the rightmost point on the horizontal line passing through the center. There are variations, like using the midpoint of OW, but the principle remains: a clever use of intersecting circles to pinpoint the vertices.

Euclid's method

Euclid himself detailed how to construct a regular pentagon in his Elements. It involves the use of the golden triangle and is a classic example of constructible polygons.

Physical construction methods

• Paper strip knot: Take a strip of paper, tie a simple overhand knot, and flatten it carefully. You’ll get a regular pentagon. It’s surprisingly effective. And if you hold it up to the light, you might even see a pentagram.
• Pyramid base: Construct a regular hexagon on cardstock. Cut from one vertex to the center, creating a flap. Fold this flap under the adjacent section to form a pentagonal pyramid. The base is your pentagon.

Symmetry

The regular pentagon possesses Dih₅ symmetry, which means it has an order of 10. Since 5 is a prime number, its symmetry subgroups are limited: one dihedral group (D₁) and two cyclic groups (Z₅ and Z₁). John Conway categorizes these symmetries, with r10 representing the full symmetry and 'a1' representing no symmetry. The dihedral symmetries are further classified based on whether their reflection lines pass through vertices (d) or edges (p). This inherent symmetry dictates how irregular forms might deviate, with only the g5 subgroup allowing for directed edges while maintaining a type of symmetry.

Regular pentagram

The pentagram, or {5/2} in Schläfli symbol notation, is the self-intersecting cousin of the regular pentagon. Its sides are the diagonals of a regular convex pentagon, and their lengths are related by the golden ratio.

Equilateral pentagons

An equilateral pentagon has five sides of equal length, but its angles can vary, leading to a family of shapes rather than a single unique one. This is in contrast to the regular pentagon, which is both equilateral and equiangular.

Cyclic pentagons

A cyclic pentagon is one whose vertices all lie on a single circle. While the regular pentagon is a prime example, other cyclic pentagons exist. The area of such a pentagon can be determined by solving a septic equation, a polynomial of degree seven. There’s a particular subset known as Robbins pentagons, which have rational sides and rational area. It's conjectured that their diagonals are also rational, though proving this is a challenge.

General convex pentagons

For any convex pentagon with side lengths $a, b, c, d, e$ and diagonals $d_1, d_2, d_3, d_4, d_5$, a general inequality holds:

$3(a^2+b^2+c^2+d^2+e^2) > d_1^2+d_2^2+d_3^2+d_4^2+d_5^2$

It’s a statement about the relationship between the sides and the diagonals, suggesting that the sum of the squares of the sides is notably larger than the sum of the squares of the diagonals.

Pentagons in tiling

The regular pentagon presents a unique challenge when it comes to tiling a plane without gaps or overlaps.

• No regular tiling: A regular pentagon cannot form a regular tiling because 360° divided by its internal angle of 108° is not an integer (360°/108° = 3⅓). This means you can't arrange a whole number of regular pentagons around a single point without leaving gaps or overlapping.
• Edge-to-edge tiling: Proving that a pentagon can't be part of any edge-to-edge tiling using regular polygons is more complex. The reasoning involves the odd number of sides, which forces alternating polygons to be congruent, a condition that doesn't work out for pentagons.
• Maximum packing density: The densest known packing of regular pentagons is a double lattice structure, achieving a density of about 92.131%. This "pentagonal ice-ray" design, known for centuries, has been proven to be the optimal packing density.
• Monohedral tiling: While regular pentagons can't tile the plane on their own, there are 15 distinct classes of pentagons that can tile the plane monohedrally (meaning only one shape of pentagon is used). These tiling pentagons, however, generally lack symmetry, though some have mirror symmetry in special cases.

Pentagons in polyhedra

The pentagon is a fundamental building block in several polyhedra, particularly those with high degrees of symmetry. Examples include the dodecahedron (which has 12 regular pentagonal faces), the pyritohedron, the tetartoid, the pentagonal icositetrahedron, and the pentagonal hexecontahedron.

Pentagons in nature

The five-sided figure appears surprisingly often in the natural world, suggesting some underlying principle at play.

• Plants: You can see pentagonal cross-sections in okra. Many flowers, like morning glories, exhibit a pentagonal symmetry in their petals. The gynoecium of an apple has five carpels arranged in a star pattern, and starfruit is another well-known example of fivefold symmetry. The perigone tube of the Rafflesia flower also displays this form.
• Animals: The fivefold radial symmetry is a hallmark of many echinoderms, such as sea stars, sea urchins, and brittle stars.
• Minerals: While often associated with perfect shapes, some minerals exhibit pentagonal faces. Ho-Mg-Zn quasicrystals can form as pentagonal dodecahedra, and pyrite crystals can have faces that are pentagonal, though not necessarily regular. A "Fiveling" of gold is another example.

Other examples

• The Pentagon: The iconic headquarters of the United States Department of Defense.
• Home plate: In baseball, the shape defining the scoring area is a pentagon.

This covers the essential, and frankly, slightly tedious, details. Anything else, and you’re on your own.