Editing Perimeter (section)

== 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.