View source for Perimeter

{{Short description|Path that surrounds an area}}
{{Other uses}}
{{pp-semi-indef}}

[[File:Perimiters.svg|thumb|250px|Perimeter is the distance around a two dimensional shape, a measurement of the distance around something; the length of the boundary.]]

A '''perimeter''' is a closed [[path (geometry)|path]] that encompasses, surrounds, or outlines either a [[two dimensional]] [[shape]] or a [[one-dimensional]] [[length (mathematics)|length]]. The perimeter of a [[circle]] or an [[ellipse]] is called its [[circumference]].

Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden. The perimeter of a wheel/circle (its circumference) describes how far it will roll in one [[revolution (geometry)|revolution]]. Similarly, the amount of string wound around a spool is related to the spool's perimeter; if the length of the string was exact, it would equal the perimeter.

== Formulas ==
{| class="wikitable sortable mw-collapsible"
|+
! shape !! formula || variables
|-
| [[circle]] || <math>2 \pi r = \pi d</math> || where <math>r</math> is the radius of the circle and <math>d</math> is the diameter.
|-
| [[semicircle]] || <math>(\pi+2)r </math>|| where <math>r</math> is the radius of the semicircle.
|-
| [[triangle]] || <math>a + b + c\,</math> || where <math>a</math>, <math>b</math> and <math>c</math> are the lengths of the sides of the triangle.
|- 
| [[square (geometry)|square]]/[[rhombus]] || <math>4a</math> || where <math>a</math> is the side length.
|- 
| [[rectangle]] || <math>2(l+w)</math> || where <math>l</math> is the length and <math>w</math> is the width.
|-
| [[equilateral polygon]] || <math>n \times a\,</math> || where <math>n</math> is the number of sides and <math>a</math> is the length of one of the sides.
|-
| [[regular polygon]] || <math>2nb \sin\left(\frac{\pi}{n}\right)</math> || where <math>n</math> is the number of sides and <math>b</math> is the distance between center of the polygon and one of the [[Vertex (geometry)|vertices]] of the polygon.
|-
| general [[polygon]] || <math>a_1 + a_2 + a_3 + \cdots + a_n = \sum_{i=1}^n a_i</math> || where <math>a_{i}</math> is the length of the <math>i</math>-th (1st, 2nd, 3rd ... ''n''th) side of an ''n''-sided polygon.
|}
[[File:Herzkurve2.svg|thumb|upright=1.0|[[cardioid]] <math>\gamma:[0,2\pi] \to \mathbb{R}^2 </math><br/>(drawing with <math>a=1</math>)<br/><math>x(t) = 2 a \cos(t) (1 + \cos(t))</math><br/><math>y(t) =  2 a \sin(t) (1 + \cos (t))</math><br/><math>L = \int_0^{2\pi} \sqrt{x'(t)^2+y'(t)^2}\,\mathrm dt = 16a</math>]]
The perimeter is the distance around a shape. Perimeters for more general shapes can be calculated, [[Arc length#Finding arc lengths by integrating|as any path]], with <math display="inline">\int_0^L \mathrm{d}s</math>, where <math>L</math> is the length of the path and <math>ds</math> is an infinitesimal line element. Both of these must be replaced by algebraic forms in order to be practically calculated. If the perimeter is given as a closed [[plane curve|piecewise smooth plane curve]] <math> \gamma: [a,b] \to \mathbb{R}^2</math> with
:<math> \gamma(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatrix}</math> 
then its length <math>L</math> can be computed as follows:
: <math>L = \int_a^b \sqrt{x'(t)^2+y'(t)^2}\,\mathrm dt</math>

A generalized notion of perimeter, which includes [[hypersurface]]s bounding volumes in <math>n</math>-[[Dimension (mathematics)|dimensional]] [[Euclidean space]]s, is described by the theory of [[Caccioppoli set]]s.

==Polygons==
[[File:PerimeterRectangle.svg|thumb|Perimeter of a rectangle.]]
[[Polygon]]s are fundamental to determining perimeters, not only because they are the simplest shapes but also because the perimeters of many shapes are calculated by [[Approximation#Mathematics|approximating]] them with [[limit of a sequence|sequences]] of polygons tending to these shapes. The first mathematician known to have used this kind of reasoning is [[Archimedes]], who approximated the perimeter of a circle by surrounding it with [[regular polygon]]s.{{r|archimedes}}

The perimeter of a polygon equals the [[summation|sum]] of the lengths of its [[Edge (geometry)|sides (edges)]]. In particular, the perimeter of a [[rectangle]] of width <math>w</math> and length <math>\ell</math> equals <math>2w + 2\ell.</math>

An [[equilateral polygon]] is a polygon which has all sides of the same length (for example, a [[rhombus]] is a 4-sided equilateral polygon). To calculate the perimeter of an equilateral polygon, one must multiply the common length of the sides by the number of sides.

A [[regular polygon]] may be characterized by the number of its sides and by its [[circumradius]], that is to say, the constant distance between its [[Centre (geometry)|centre]] and each of its [[Vertex (geometry)|vertices]]. The length of its sides can be calculated using [[trigonometry]]. If {{math|''R''}} is a regular polygon's radius and {{math|''n''}} is the number of its sides, then its perimeter is 
:<math>2nR \sin\left(\frac{180^{\circ}}{n}\right).</math>

A [[splitter (geometry)|splitter]] of a [[triangle]] is a [[cevian]] (a segment from a vertex to the opposite side) that divides the perimeter into two equal lengths, this common length being called the [[semiperimeter]] of the triangle. The three splitters of a triangle [[concurrent lines|all intersect each other]] at the [[Nagel point]] of the triangle.

A [[cleaver (geometry)|cleaver]] of a triangle is a segment from the midpoint of a side of a triangle to the opposite side such that the perimeter is divided into two equal lengths. The three cleavers of a triangle all intersect each other at the triangle's [[Spieker center]].

==Circumference of a circle==
[[File:Pi-unrolled-720.gif|right|300px|thumb|If the diameter of a circle is 1, its circumference equals {{pi}}.]]
{{Main|Circumference}}
The perimeter of a [[circle]], often called the circumference, is proportional to its [[diameter]] and its [[radius]]. That is to say, there exists a constant number [[pi]], {{pi}} (the [[ancient greek|Greek]] ''p'' for perimeter), such that if {{math|''P''}} is the circle's perimeter and {{math|''D''}} its diameter then,
:<math>P = \pi\cdot{D}.\!</math>

In terms of the radius {{math|''r''}} of the circle, this formula becomes,

:<math>P=2\pi\cdot r.</math>

To calculate a circle's perimeter, knowledge of its radius or diameter and the number {{pi}} suffices. The problem is that {{pi}} is not [[rational number|rational]] (it cannot be expressed as the [[quotient]] of two [[integer]]s), nor is it [[algebraic number|algebraic]] (it is not a root of a polynomial equation with rational coefficients). So, obtaining an accurate approximation of {{pi}} is important in the calculation. The computation of the digits of {{pi}} is relevant to many fields, such as [[mathematical analysis]], [[algorithmics]] and [[computer science]].

== Perception of perimeter==
{{multiple image
 | image1 = Hexaflake.gif
 | caption1 = The more one cuts this shape, the lesser the area and the greater the perimeter. The [[convex hull]] remains the same.
 | image2 = Neuf Brisach.jpg
 | caption2 = The [[Neuf-Brisach]] fortification perimeter is complicated. The shortest path around it is along its [[convex hull]].
 | total_width = 400
 | align = right
}}
{{Main|Area (geometry)|convex hull}}
The perimeter and the [[area (geometry)|area]] are two main measures of geometric figures. Confusing them is a common error, as well as believing that the greater one of them is, the greater the other must be. Indeed, a commonplace observation is that an enlargement (or a reduction) of a shape make its area grow (or decrease) as well as its perimeter. For example, if a field is drawn on a 1/{{formatnum:10000}} scale map, the actual field perimeter can be calculated multiplying the drawing perimeter by {{formatnum:10000}}. The real area is {{formatnum:10000}}{{sup|2}} times the area of the shape on the map. Nevertheless, there is no relation between the area and the perimeter of an ordinary shape. For example, the perimeter of a rectangle of width 0.001 and length 1000 is slightly above 2000, while the perimeter of a rectangle of width 0.5 and length 2 is 5. Both areas are equal to 1.

[[Proclus]] (5th century) reported that Greek peasants "fairly" parted fields relying on their perimeters.{{r|heath}} However, a field's production is proportional to its area, not to its perimeter, so many naive peasants may have gotten fields with long perimeters but small areas (thus, few crops).

If one removes a piece from a figure, its area decreases but its perimeter may not. The convex hull of a figure may be visualized as the shape formed by a rubber band stretched around it.{{r|convexhull}} In the animated picture on the left, all the figures have the same convex hull; the big, first [[hexagon]].

== Isoperimetry ==
{{Further|Isoperimetric inequality}}

The isoperimetric problem is to determine a figure with the largest area, amongst those having a given perimeter. The solution is intuitive; it is the [[circle]]. In particular, this can be used to explain why drops of fat on a [[broth]] surface are circular.

This problem may seem simple, but its mathematical proof requires some sophisticated theorems. The isoperimetric problem is sometimes simplified by restricting the type of figures to be used. In particular, to find the [[quadrilateral]], or the triangle, or another particular figure, with the largest area amongst those with the same shape having a given perimeter. The solution to the quadrilateral isoperimetric problem is the [[square]], and the solution to the triangle problem is the [[equilateral triangle]]. In general, the polygon with {{math|''n''}} sides having the largest area and a given perimeter is the [[regular polygon]], which is closer to being a circle than is any irregular polygon with the same number of sides.

==Etymology==
The word comes from the [[Ancient Greek|Greek]] περίμετρος ''perimetros'', from περί ''peri'' "around" and μέτρον ''metron'' "measure".

== See also ==
{{div col|colwidth=23em}}
* [[Arclength]]
* [[Area]]
* [[Coastline paradox]]
* [[Girth (geometry)]]
* [[Pythagorean theorem]]
* [[Surface area]]
* [[Volume]]
* [[Wetted perimeter]]
{{div col end}}

==References==
{{Reflist|refs=

<ref name="archimedes">{{cite book
 | last1 = Varberg | first1 = Dale E.
 | last2 = Purcell | first2 = Edwin J.
 | last3 = Rigdon | first3 = Steven E.
 | title = Calculus
 | year = 2007
 | publisher = [[Pearson Prentice Hall]]
 | edition = 9th
 | isbn = 978-0131469686
 | page = 215&ndash;216
}}</ref>

<ref name="convexhull">{{cite book
 | last1 = de Berg | first1 = M. | author1-link = Mark de Berg
 | last2 = van Kreveld | first2 = M. | author2-link = Marc van Kreveld
 | last3 = Overmars | first3 = Mark | author3-link = Mark Overmars
 | last4 = Schwarzkopf | first4 = O. | author4-link = Otfried Cheong
 | edition = 3rd
 | publisher = Springer
 | title = Computational Geometry: Algorithms and Applications
 | year = 2008
 | page = 3
}}</ref>

<ref name="heath">{{cite book
 | last = Heath | first = T.
 | title = A History of Greek Mathematics
 | volume = 2
 | publisher = [[Dover Publications]]
 | year = 1981
 | page = 206
 | isbn = 0-486-24074-6
}}</ref>

}}

== External links ==
{{Wiktionary}}
{{Wikibooks|Geometry|Chapter 8|Perimeters, areas and volumes}}
{{Wikibooks|Geometry|Perimeter and Arclength}}
{{Wikibooks|Geometry|Circles/Arcs|Arcs}}
* {{MathWorld |urlname=Perimeter |title=Perimeter}}

{{Authority control}}

[[Category:Elementary geometry]]
[[Category:Length]]