Perimeter - Sarcasm Wiki

Contents

1. Overview
2. Etymology
3. Cultural Impact

For other uses, see Perimeter (disambiguation) .

The perimeter is, to put it simply, the distance that encircles a two-dimensional shape —the measurable extent of its boundary . It’s the total length of the line segments or curves that define the outer edge of a figure. When discussing the perimeter of a circle or an ellipse , this specific measurement is more precisely referred to as its circumference . Essentially, a perimeter quantifies the complete length of a closed boundary that either encloses, surrounds, or delineates a two-dimensional shape or, in a slightly more abstract sense, a one-dimensional line .

The calculation of a perimeter isn’t merely an academic exercise; it carries several profoundly practical implications. Consider the mundane task of enclosing a yard or garden: the calculated perimeter directly dictates the precise length of fence material required. Or, observe a wheel in motion: its perimeter, or circumference , directly describes the distance it will traverse in a single complete revolution . Similarly, the total amount of string meticulously wound around a spool is intrinsically linked to the spool’s perimeter; if the string length were to be exact, it would precisely correspond to that perimeter. These are not profound mysteries, merely the fundamental truths of spatial measurement that you, presumably, are attempting to grasp.

Formulas

To save us both time, here are the conventional formulas for calculating the perimeter of common geometric entities. Don’t expect these to solve all your problems, but they’re a start.

| shape | formula
Circle (Circumference) | $2\pi r = \pi d$ | where $r$ is the radius of the circle and $d$ is the diameter. Semicircle | $(\pi + 2)r$ | where $r$ is the radius of the semicircle. Triangle | $a+b+c$ | where $a$, $b$, and $c$ are the lengths of the sides of the triangle. Square /Rhombus | $4a$ | where $a$ is the side length. Rectangle | $2(l+w)$ | where $l$ is the length and $w$ is the width. Equilateral polygon | $n \times a$ | where $n$ is the number of sides and $a$ is the length of one of the sides. Regular polygon | $2nb \sin \left({\frac {\pi }{n}}\right)$ | where $n$ is the number of sides and $b$ is the distance between the center of the polygon and one of its vertices . General polygon | $a_1+a_2+a_3+\cdots+a_n = \sum {i=1}^{n}a{i}$ | where $a_i$ is the length of the $i$-th (1st, 2nd, 3rd … $n$-th) side of an $n$-sided polygon. Cardioid | $L = \int _{0}^{2\pi }{\sqrt {x’(t)^{2}+y’(t)^{2}}},\mathrm {d} t=16a$ | Given by the parametric equations $\gamma :[0,2\pi ]\to \mathbb {R} ^{2}$ as $x(t)=2a\cos(t)(1+\cos(t))$ and $y(t)=2a\sin(t)(1+\cos(t))$, (drawing with $a=1$).

The perimeter represents the total distance around a shape . For more complex or general shapes, the perimeter can be calculated as an arc length . This involves integrating an infinitesimal line element along the entire path that defines the boundary. Specifically, the integral form is $\int _{0}^{L}\mathrm{d}s$, where $L$ is the total length of the path and $ds$ represents an infinitesimally small segment of that line. For practical computation, these abstract elements must be converted into algebraic forms.

If the perimeter is defined by a closed piecewise smooth plane curve $\gamma :[a,b]\to \mathbb {R} ^{2}$, where the curve’s points are given by $\gamma (t)={\begin{pmatrix}x(t)\y(t)\end{pmatrix}}$, then its total length , $L$, can be precisely computed using the following integral:

$L=\int _{a}^{b}{\sqrt {x’(t)^{2}+y’(t)^{2}}},\mathrm {d} t$

This formula essentially sums up the lengths of infinitely small segments along the curve, where $x’(t)$ and $y’(t)$ are the derivatives of the parametric functions with respect to $t$, representing the instantaneous rates of change of the x and y coordinates.

A more generalized and sophisticated concept of perimeter exists, extending to hypersurfaces that bound volumes within $n$-dimensional Euclidean spaces . This advanced notion is rigorously described by the theory of Caccioppoli sets , which provides a framework for defining and measuring boundaries in higher dimensions, often in contexts where classical differentiable manifolds are insufficient. It’s an area far beyond what you’d typically need for a garden fence, thankfully.

Polygons

Polygons are undeniably fundamental in the realm of perimeter determination. This isn’t merely because they are the simplest geometric constructs, but rather because the perimeters of a vast array of more complex shapes are often derived by approximating them with judiciously chosen sequences of polygons that progressively “tend” towards the ultimate, desired shape. The earliest known mathematician to master and employ this ingenious method was Archimedes . He famously approximated the perimeter, or circumference , of a circle by inscribing and circumscribing it with regular polygons of increasing numbers of sides. This pioneering approach laid foundational groundwork for integral calculus, demonstrating how an infinite process could yield a precise measurement for a curved boundary.

For any polygon , regardless of its complexity or regularity, its perimeter is straightforwardly calculated as the sum of the lengths of all its individual sides (edges) . In the particular case of a rectangle with a specific width, denoted as $w$, and a distinct length, $\ell$, its perimeter is elegantly expressed as $2w+2\ell$. This formula arises from the fact that a rectangle possesses two pairs of equal-length sides.

An equilateral polygon is defined by the characteristic that all of its sides possess precisely the same length . A classic example of a 4-sided equilateral polygon is a rhombus . To compute the perimeter of any equilateral polygon , one simply multiplies the common length of its sides by the total number of sides it possesses. This provides a simplified calculation due to the inherent symmetry of its side lengths.

A regular polygon exhibits an even higher degree of symmetry, characterized not only by equal side lengths but also by equal interior angles. Such a polygon can be uniquely defined by its number of sides, $n$, and its circumradius , $R$. The circumradius is the constant distance from the polygon’s centre to each of its vertices . The length of its sides, and consequently its perimeter, can be precisely calculated using fundamental principles of trigonometry . If $R$ denotes the circumradius of a regular polygon and $n$ represents the number of its sides, then its perimeter is given by the formula:

$2nR\sin \left({\frac {180^{\circ }}{n}}\right).$

This formula relies on dividing the polygon into $n$ congruent isosceles triangles with their apexes at the center and bases as the polygon’s sides.

In the specific context of a triangle , certain specialized segments are defined based on their interaction with the perimeter. A splitter of a triangle is a particular type of cevian —a line segment extending from a vertex to the opposite side—that has the unique property of dividing the triangle’s perimeter into two segments of equal length . This common length is often referred to as the semiperimeter of the triangle. Intriguingly, all three splitters of any given triangle are concurrent , meaning they invariably intersect each other at a single, unique point known as the Nagel point of the triangle.

Similarly, a cleaver of a triangle is a segment that originates from the midpoint of one of the triangle’s sides and extends to the opposite side, also dividing the perimeter into two equal lengths . Just like splitters , the three cleavers of a triangle are also concurrent , converging at a distinct point called the Spieker center , which is the incenter of the medial triangle . These geometric curiosities illustrate the nuanced ways in which perimeter can be subdivided and analyzed within even seemingly simple shapes.

Circumference of a circle

Main article: Circumference

The perimeter of a circle , a measurement so fundamental it has its own distinct name—the circumference —is universally proportional to both its diameter and its radius . This inherent relationship is governed by an extraordinary and ubiquitous mathematical constant known as pi , denoted by the Greek letter π (a historical nod, as ‘p’ once stood for perimeter). Consequently, if $P$ represents the circle’s perimeter and $D$ its diameter , their relationship is elegantly expressed as:

$P = \pi \cdot D.$

Alternatively, when expressed in terms of the circle’s radius , $r$, this formula becomes:

$P = 2\pi \cdot r.$

To accurately calculate a circle’s perimeter, one requires precise knowledge of either its radius or diameter , and, crucially, the value of the number π. This is where things become… interesting. The inherent challenge lies in the nature of π itself: it is not a rational number , meaning it cannot be expressed as a simple quotient of two integers . More profoundly, π is not merely irrational but also transcendental (a subtle but critical distinction from being merely algebraic; it is not a root of any non-zero polynomial equation with rational coefficients). This means that any decimal representation of π is infinitely long and non-repeating, rendering an exact numerical value impossible to write down.

Therefore, obtaining an accurate approximation of π becomes paramount for practical calculations. The ongoing computational pursuit of ever more digits of π, while seemingly esoteric, holds significant relevance across numerous scientific and mathematical disciplines. These efforts contribute not only to fields such as mathematical analysis and algorithmics but also push the boundaries of computer science in terms of computational efficiency and precision. It’s a testament to humanity’s persistent, if sometimes futile, quest for ultimate exactitude.

Perception of perimeter

Main articles: Area (geometry) and convex hull

The perimeter and the area are two of the most fundamental quantitative measures used to describe geometric figures . Despite their common association, confusing these two distinct concepts is a remarkably frequent error, as is the misguided belief that an increase in one must necessarily correspond to an increase in the other. While it’s true that a uniform enlargement or reduction of a shape will cause both its area and its perimeter to grow or shrink proportionally (for instance, if a field is represented on a 1/10,000 scale map, its actual perimeter is 10,000 times the drawing’s perimeter, and its actual area is $10,000^2$ times the map’s area), this proportional relationship does not hold universally.

Indeed, there exists no inherent or direct relationship between the area and the perimeter of an arbitrary shape . Consider, for example, two rectangles : one with an extraordinarily small width of 0.001 units and an immense length of 1000 units. Its perimeter would be approximately $2(1000 + 0.001) \approx 2000.002$ units. Now, imagine another rectangle with a width of 0.5 units and a length of 2 units. Its perimeter is $2(2 + 0.5) = 5$ units. Both of these vastly different rectangles, however, possess an identical area of 1 square unit ($1000 \times 0.001 = 1$ and $2 \times 0.5 = 1$). This stark contrast perfectly illustrates that a large perimeter does not guarantee a large area, nor does a small area imply a small perimeter.

The ancient Greek philosopher Proclus (5th century AD) recorded a telling anecdote: Greek peasants, he noted, would “fairly” partition agricultural fields based solely on their perimeters . This, of course, was a catastrophic misjudgment. A field’s agricultural yield is directly proportional to its area , not the length of its boundary. Consequently, many naive peasants likely ended up with parcels of land boasting impressively long perimeters but meager areas , resulting in distressingly few crops. A classic example of human folly, perhaps.

Furthermore, if one were to remove a section from a figure , its area would invariably decrease. However, counterintuitively, its perimeter might not only remain the same but could even increase, depending on the nature of the cut. This phenomenon becomes clearer when considering the concept of a convex hull . The convex hull of a figure can be intuitively visualized as the shape that would be formed by a taut rubber band stretched around its outer extremities. In the animated image provided, all the depicted figures, despite their internal complexities, share the exact same convex hull —the large, initial hexagon . The internal cuts and indentations reduce the area but often extend the actual boundary length , making the perimeter grow while the convex hull remains unchanged. Similarly, the Neuf-Brisach fortification, with its intricate star-shaped design, presents a highly complicated perimeter. The shortest path around it, however, is simply the length of its convex hull , which smooths out all the defensive intricacies.

Isoperimetry

Further information: Isoperimetric inequality

The isoperimetric problem is a classic mathematical challenge: to identify the figure that encloses the greatest possible area from among all those possessing a given, fixed perimeter . The solution, while intuitively appealing, requires surprisingly sophisticated mathematical proof: the circle . This fundamental principle explains various natural phenomena, such as why drops of fat, oil, or water on a surface tend to assume a perfectly circular shape —it is the most efficient configuration for a given amount of ‘boundary’ to enclose the maximum ‘substance’, minimizing surface tension.

While the core isoperimetric problem might seem conceptually simple, its rigorous mathematical proof necessitates advanced theorems and techniques, often rooted in the calculus of variations . For practical or pedagogical reasons, the isoperimetric problem is frequently simplified by restricting the class of figures under consideration. For instance, one might seek to find the quadrilateral , or the triangle , or another specific type of polygon , that yields the largest area for a predefined perimeter while maintaining its particular shape .

For quadrilaterals, the solution to this restricted isoperimetric problem is invariably the square . Among all triangles with a fixed perimeter , the equilateral triangle encloses the largest area . Generalizing this, for any given number of sides, $n$, the polygon that possesses the largest area for a specified perimeter is always the regular polygon with $n$ sides. This is because regular polygons are the most symmetrical and “circle-like” among polygons of the same side count, thus approaching the efficiency of the ultimate solution, the circle . The more sides a regular polygon has, the closer its shape and area-to-perimeter ratio will approximate that of a circle .

Etymology

The word “perimeter” originates from the Ancient Greek term περίμετρος (perimetros). This word is a compound of two distinct elements: περί (peri), meaning “around” or “about,” and μέτρον (metron), which translates to “measure.” Thus, the term literally means “a measure around,” succinctly encapsulating its geometric definition.