1,183 Pages In any triangle, the distance around the boundary of the triangle from a vertex to the point on the opposite edge touched by an excircle equals the semiperimeter.

In geometry, the semiperimeter of a polygon is half its perimeter Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name. When the semiperimeter occurs as part of a formula, it is typically denoted by the letter s.

The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths a, b, and c is:   The area of any triangle is the product of its inradius and its semiperimeter; the same area formula also applies to tangential quadrilaterals, in which pairs of opposite sides have lengths adding to the semiperimeter. The area of a triangle can also be calculated from its semiperimeter and side lengths using Heron's formula: The simplest form of Brahmagupta's formula, for the area of a cyclic quadrilateral, has a similar form: The circumradius R of a triangle can also be calculated from the semiperimeter and side lengths: This formula can be derived from the law of sines.  