What is a triangle circumcircle?
The circumscribed circle, or circumcircle, of a triangle is the unique circle that passes through all three of its vertices. Its center is the circumcenter (the point where the three perpendicular bisectors of the sides meet), and its radius is called the circumradius. This calculator finds the circumradius, diameter and circumcircle area directly from the three side lengths, and also reports the triangle's own area and the ratio between the two areas.
How to use it
Enter the three side lengths a, b and c in any consistent length unit (all in cm, all in m, etc.). The tool returns the radius and diameter in that same unit, and both areas in that unit squared. The sides must be positive and must form a valid triangle: the sum of any two sides must be greater than the third. If they do not, the result is flagged as invalid.
The formula explained
First compute the semi-perimeter \(s = (a + b + c) / 2\). Heron's formula then gives the triangle area \(S_t = \sqrt{s(s-a)(s-b)(s-c)}\). The circumradius follows from the identity $$r = \frac{a\,b\,c}{4\,S_t}.$$ From \(r\) we get the diameter \(\phi = 2r\), the circumcircle area \(S_c = \pi r^2\), and finally the area ratio \(S_c / S_t\).
Worked example
Take the 3-4-5 right triangle. The semi-perimeter is \(s = 6\), so $$S_t = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6.$$ The circumradius is $$r = \frac{3 \times 4 \times 5}{4 \times 6} = \frac{60}{24} = 2.5,$$ the diameter is 5, the circumcircle area is \(\pi \times 2.5^2 = 19.6350\), and the ratio \(S_c / S_t = 3.2725\). Note the diameter equals the hypotenuse 5 — a known property of right triangles.
FAQ
Why is my triangle invalid? Either a side is zero or negative, or the longest side is at least as long as the other two combined, which cannot form a closed triangle.
What units does the result use? Whatever unit you used for the sides: the radius and diameter share that unit, and areas are in that unit squared.
Does it work for any triangle? Yes — acute, right or obtuse — as long as it is a genuine non-degenerate triangle.
