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[[File:Perimiters.svg|thumb|250px|Perimeter is the distance around a two dimensional shape, or the measurement of the distance around something; the length of the boundary.]]
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A '''perimeter''' is a path that surrounds a [[two-dimensional]] [[shape]]. The word comes from the Greek ''peri'' (around) and ''meter'' (measure). The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a [[circle]] or [[ellipse]] is called its [[circumference]].
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== Formulae ==
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{| class="wikitable"
! 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.
|-
| [[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.
|}


The perimeter is the distance around all of a shape. Perimeters for more general shapes can be calculated as any path with <math>\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 with other algebraic forms in order to be solved: an advanced notion of perimeter, which includes [[hypersurface]]s bounding volumes in <math>n</math>-[[Dimension (mathematics)|dimensional]] [[Euclidean space]]s can be found in the theory of [[Caccioppoli set]]s.
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[[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.


The perimeter of a polygon equals the [[sum]] of the lengths of its [[Edge (geometry)|edges]]. In particular, the perimeter of a [[rectangle]] which width is <math>w</math> and length <math>\ell</math> is equal to <math>2w + 2\ell</math>.
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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.
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A [[regular polygon]] may be defined by the number of its sides and by its ''radius'', that is to say, the constant distance between its [[Centre (geometry)|centre]] and each of its [[Vertex (geometry)|vertices]]. One can calculate the length of its sides using [[trigonometry]]. If ''R'' is a regular polygon's radius and ''n'' is the number of its sides, then its perimeter is
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==Circumference of a circle==
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[[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 ''P'' is the circle's perimeter and ''D'' its diameter then:
:<math>P = \pi\cdot{D}.\!</math>


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To calculate a circle's perimeter, knowledge of its radius or diameter and of the number {{pi}} is sufficient. 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 for the calculation. The search for the digits of {{pi}} is relevant to many fields, such as [[mathematical analysis]], [[algorithmics]] and [[computer science]].
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[[File:Hexaflake.gif|thumb|left|upright=0.6|The more one cuts this shape, the lesser the area and the greater the perimeter. The [[convex hull]] remains the same.]]
[[File:Neuf Brisach.jpg|thumb|The [[Neuf-Brisach]] fortification perimeter is complicated. The shortest path around it is along its [[convex hull]].]]
{{Main|Area (geometry)|convex hull}}
The perimeter and the [[area (geometry)|area]] are the main two measures of geometric figures. Confusing them is frequent, as well as believing that the greater one of them is, the greater is the other. Indeed, 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 equal to 1.
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[[Proclus]] (5th century) reported that Greek peasants "fairly" parted fields relying on their perimeters.<ref>{{cite book|first1=T.|last1=Heath|title=A History of Greek Mathematics|volume=2|publisher=Dover Publications|year= 1981|page= 206|isbn=0-486-24074-6}}</ref> But a field's production is proportional to its area, not to its perimeter: many naive peasants may have got fields with long perimeters but low areas (thus, low crops).
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If one removes a piece from a figure, its area decreases but its perimeter may not. In the case of very irregular shapes, some people may confuse perimeter with [[convex hull]]. The convex hull of a figure may be visualized as the shape formed by a rubber band stretched around it. On the animated picture on the left, all the figures have the same convex hull: the big, first [[hexagon]].
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== Isoperimetry ==
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{{Further|Isoperimetric inequality}}
 
 
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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, that is why drops of fat on a [[broth]] surface are circular.
 
 
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This problem may seem simple, but its mathematical proof needs sophisticated theorems. The isoperimetric problem is sometimes simplified: to find the [[quadrilateral]], or the triangle or another particular figure, with the largest area amongst those 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 ''n'' sides having the largest area and a given perimeter is the [[regular polygon]], which is closer to being a circle than is an irregular polygon.
 
 
</ul>
== See also ==
* [[Arclength]]
* [[Area]]
* [[Caccioppoli set]]
* [[Circumference]]
* [[Isoperimetric inequality]]
* [[Pythagorean theorem]]
* [[Volume]]
* [[Wetted perimeter]]
 
==References==
<references/>
 
== External links ==
{{wiktionary}}
{{wikibooks|Geometry|Chapter 8|Perimeters, areas and volumes}}
{{wikibooks|Geometry|Perimeter and Arclength}}
{{wikibooks|Geometry|Circles/Arcs|Arcs}}
{{Translation/Ref|fr|Pétimètre|oldid=85487061}}
 
* {{MathWorld |urlname=Perimeter |title=Perimeter}}
* {{MathWorld |urlname=Semiperimeter |title=Semiperimeter}}
 
[[Category:Elementary geometry]]
[[Category:Length]]
 
{{Link GA|fr}}

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