The sides of a triangle are 13 cm, 14 cm and 15 cm. What is the radius (in cm) of the circle that passes through all three vertices of the triangle?

The required circle is the circumcircle, with radius R = abc/(4 × Area).

Area by Heron's formula: s = (13 + 14 + 15)/2 = 21.
Area = √(21 × 8 × 7 × 6) = √7056 = 84 cm².

R = (13 × 14 × 15)/(4 × 84) = 2730/336 = 65/8 = 8.125 cm.

Tip: The 13-14-15 triangle is two right triangles (5-12-13 and 9-12-15) joined along the altitude 12, which gives the area 84 instantly. (Its inradius is Area/s = 4 — a common confusion.)