The formula
r = area ÷ s · R = abc ÷ (4 × area)
Where it comes from
Every triangle has exactly one circle that fits inside touching all three sides, and exactly one that passes through all three vertices. Both radii follow from the sides alone, by way of the area.
How to work it out by hand
- Work out the semi-perimeter and the area by Heron's formula
- Inradius: the area divided by the semi-perimeter
- Circumradius: the three sides multiplied, divided by four times the area
- Multiply either by 2π for its circumference
What is worth knowing
Euler found a clean relationship between the two: the distance d between the centres satisfies d² = R(R − 2R·r/R), which forces R to be at least twice r for any triangle. Equality holds only for the equilateral triangle, where both centres coincide — so the ratio you get here is a direct measure of how far from equilateral your triangle is. For a right triangle there is a much simpler fact worth remembering: the circumradius is exactly half the hypotenuse, because the hypotenuse is a diameter of the circumcircle. That is Thales' theorem, and it is why any angle inscribed in a semicircle is a right angle.