How this calculator works
First the tool checks the triangle inequality and finds the triangle area. The circumradius follows from the product of the three sides divided by four times that area. The circumcentre can lie outside an obtuse triangle; the circle still passes through every vertex. The inradius is the radius of a different circle, tangent to all three sides.
Formula
s = (a + b + c)/2; K = √[s(s-a)(s-b)(s-c)]; R = abc/(4K); r = K/s; circle area = πR².
Worked example
A 3 m, 4 m, 5 m triangle has area 6 m². Its circumradius is 60/24 = 2.5 m, its circle area is 6.25π m² and its inradius is 1 m.
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Frequently asked questions
Does every triangle have a circumcircle?
Every non-flat triangle has exactly one. Collinear vertices do not define a finite circumcircle.
What happens for a right triangle?
Its hypotenuse is a diameter of the circumcircle, so a hypotenuse of 10 cm gives R = 5 cm.
Which way round is the area ratio?
It is triangle area divided by circle area. It is dimensionless and less than one.