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Concept
Perimeter
Summary
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional 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. 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.
The perimeter is the distance around a shape. Perimeters for more general shapes can be calculated, as any path, with , where is the length of the path and 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 piecewise smooth plane curve with
then its length can be computed as follows:
A generalized notion of perimeter, which includes hypersurfaces bounding volumes in -dimensional Euclidean spaces, is described by the theory of Caccioppoli sets.
Polygons are fundamental to determining perimeters, not only because they are the simplest shapes but also because the perimeters of many shapes are calculated by approximating them with 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 polygons.
The perimeter of a polygon equals the sum of the lengths of its sides (edges). In particular, the perimeter of a rectangle of width and length equals
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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