Editing Perimeter (section)

==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]].