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The perimeter of a polygon is the distance around it, the sum of the lengths of all its sides.

A perimeter is a path that surrounds an area. The word comes from the Greek peri (around) and meter (measure). The term may be used either for the path or its length. The perimeter of a circular area is called circumference.

## Practical uses

Calculating the perimeter has considerable practical applications. The perimeter can be used to calculate the length of fence required to surround a yard or garden. The perimeter of a wheel (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.

## Formulas

shape formula variables
circle $2 \pi r\,$ where $r$ is the radius.
triangle $a + b + c\,$ where $a$, $b$ and $c$ are the lenghts of the sides of the triangle.
equilateral polygon $n \times a\,$ where $n$ is the number of sides and $a$ is the length of one of the sides.
regular polygon $2nb \sin(\frac{\pi}{n})$ where $n$ is the number of sides and $b$ is the distance between center of the polygon and one of the vertices of the polygon.
general polygon $a_{1} + a_{2} + a_{3} + \ldots + a_{n} = \sum_{i=1}^{n}a_{i}$ where $a_{i}$ is the length of the $i$-th (1st, 2nd, 3rd ... n-th) side of an n-sided polygon.

Perimeters for more general shapes can be calculated as any path with $\int_0^Lds$ where $L$ is the length of the path and $ds$ 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 hypersurfaces bounding volumes in $n$-dimensional euclidean spaces can be found in the theory of Caccioppoli sets.