perimeters

The perimeter of a two-dimensional figure is the total length of its boundary, the distance around the figure. For a simple closed shape, it is the length traced when walking around the edge.

In polygons, the perimeter equals the sum of the lengths of its sides. For a regular polygon with n equal sides of length s, the perimeter is P = n × s. Examples include a rectangle with length l and width w, which has P = 2(l + w); a triangle with side lengths a, b, c, which has P = a + b + c; and a square with side a, which has P = 4a.

For circles and other curved shapes, the perimeter is called the circumference. It is C = 2πr, where r is the radius, or C = πd, where d is the diameter. The circumference is proportional to the circle’s size, and it is a different measure than area, though both describe aspects of the same object.

For irregular shapes, the perimeter can be found by measuring the boundary directly or by decomposing the shape into simpler parts and summing their exterior boundary lengths. If a shape has holes, the inner boundaries contribute to the total perimeter as well.

Perimeter has practical applications in fencing, track design, land surveying, and manufacturing. It is a fundamental metric used alongside area, and scaling a shape by a factor k scales its perimeter by the same factor for similar shapes.