Perimeter

For other uses, see Perimeter (disambiguation).

Perimeter is the distance around a two dimensional shape, or the measurement of the distance around something; the length of the boundary.

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 - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called its circumference.

Practical uses

The formula for the perimeter of a rectangle.

When a circle's radius is 1, its perimeter is 2π, which is also the distance it rolls in one revolution.

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 lengths of the sides of the triangle.
square/rhombus	4s	where s is the side length
rectangle	2(l + w)	where l is the length and w is the width
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 ai is the length of the i-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 $\int_0^L \mathrm{d}s$ 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.

See also

• Isoperimetric inequality
• Circumference
• Caccioppoli set
• Pythagorean Theorem
• Wetted perimeter
• Area
• Volume

External links

• Weisstein, Eric W., "Perimeter" from MathWorld.
• Weisstein, Eric W., "Semiperimeter" from MathWorld.