Inradius

An inradius is the radius of the incircle, the circle inscribed in a polygon. The incircle is tangent to every side of the polygon, and its center is equidistant from all sides. A polygon has an inradius only if such a circle exists; this occurs precisely when all internal angle bisectors meet at a single point inside the polygon.

In triangles, the inradius has a standard formula. For a triangle with side lengths a, b, c and area A, let s = (a + b + c)/2 be the semiperimeter. Then A = r s, so r = A / s. Using Heron’s formula A = sqrt[s(s−a)(s−b)(s−c)], giving r = sqrt[(s−a)(s−b)(s−c)/s].

For a general tangential polygon (one with an incircle), the area is A = r s, where s is the semiperimeter. The same radius r can be computed if A and s are known. The existence of an incircle is equivalent to the concurrency of all internal angle bisectors.

In regular polygons, the inradius has simple expressions. For a regular n-gon with side length a, the inradius is r = a/(2 tan(pi/n)). The inradius also equals the apothem, the distance from the center to any side; equivalently, if the circumradius R is known, r = R cos(pi/n).

A common special case is the tangential quadrilateral, which has an incircle if and only if opposite sides satisfy a + c = b + d (Pitot’s theorem).