exradius

An exradius is the radius of an excircle of a triangle. An excircle is a circle tangent to one side of the triangle and to the extensions of the other two sides. There are three excircles, corresponding to the three vertices, with radii r_a, r_b, and r_c, called the exradii opposite A, B, and C respectively.

The centers of the excircles are the excenters I_a, I_b, and I_c. Each excenter is formed by the intersection of the internal angle bisector at one vertex and the external angle bisectors at the other two vertices. For example, the A-excenter I_a is the intersection of the internal bisector of angle A with the external bisectors of angles B and C; the excircle opposite A is tangent to side BC and to the extensions of AB and AC.

Let a, b, c be the side lengths opposite angles A, B, C, and let s be the triangle’s semiperimeter, s = (a + b + c)/2. Let Δ denote the area of the triangle. The inradius r satisfies Δ = r s. The exradii are given by

r_a = Δ / (s − a),

r_b = Δ / (s − b),

r_c = Δ / (s − c).

Equivalently, using Heron's formula for the area Δ = sqrt[s(s − a)(s − b)(s − c)], each exradius can be expressed in terms of the sides. Exradii are generally larger than the inradius, reflecting that excircles lie outside the triangle and touch the extensions of sides. Exradii appear in various geometric formulas involving areas, tangent lengths, and angle bisectors.