Incircle & Circumcircle Radius Calculator
From the three side lengths of a triangle, find the inradius (inscribed circle) and circumradius (circumscribed circle) plus the area and perimeter.
How it works
s = (a + b + c)/2 · Area = √(s(s−a)(s−b)(s−c)) · r = Area / s · R = abc / (4·Area)
The incircle is the largest circle that fits inside the triangle, touching all three sides; its radius equals the area divided by the semi-perimeter. The circumcircle passes through all three vertices; its radius is the product of the sides over four times the area. Both follow directly from Heron’s area, so this calculator works for any valid triangle. Lengths that break the triangle inequality (one side ≥ the sum of the other two) describe no triangle and are rejected.
Worked example
For a 3-4-5 right triangle: s = 6, Area = √(6·3·2·1) = 6, r = 6/6 = 1, and R = (3·4·5)/(4·6) = 60/24 = 2.5 — exactly half the hypotenuse, as expected for a right triangle.
Sources: Incircle and excircles — Wikipedia
Frequently asked questions
Why is the circumradius half the hypotenuse for a right triangle?
In a right triangle the hypotenuse is a diameter of the circumcircle, so the circumradius is exactly half the hypotenuse. That is Thales’ theorem.
What is the triangle inequality?
For three lengths to form a triangle, each side must be shorter than the sum of the other two. If one side equals or exceeds that sum, the sides collapse to a line or cannot meet, so no triangle exists.
How are the inradius and circumradius related to the area?
The inradius is Area / s where s is the semi-perimeter, and the circumradius is abc / (4·Area). Both are derived from Heron’s formula for the area.
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This calculator is for educational purposes. Double-check important results before acting on them.