Circumscribed Circle Radius Calculator
The circumscribed circle of a triangle, or circumcircle, is the unique circle that passes through all three of its vertices. Its radius, the circumradius, is found from the three side lengths and the triangle's area. This calculator takes the three sides and returns the circumradius using the standard relationship that the circumradius equals the product of the three sides divided by four times the area. The area is computed from the sides by Heron's formula, so you never need an angle or a height. The circumcircle is centered at the circumcenter, the point where the perpendicular bisectors of the sides meet, which can lie inside, on, or outside the triangle depending on its shape. For a right triangle the circumcenter sits on the hypotenuse and the circumradius is exactly half the hypotenuse. Surveyors, machinists, drafters, geometers and students use the circumradius to fit a triangle inside a circle and to characterize triangles. Enter the three sides to get the circumradius immediately, along with the area the calculator used. All three inputs are editable so you can test any valid triangle. Every figure here is computed deterministically from the formula shown below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
A triangle's circumscribed-circle radius is abc / (4 x area). For sides 3, 4, 5 the area is 6, so the circumradius is 2.50.
Circumscribed Circle Radius formula
R = (a x b x c) / (4 x Area)
Area = sqrt( s (s-a)(s-b)(s-c) ) (Heron)
s = (a + b + c) / 2
a, b, c = the three side lengths
R = circumradius
The circumradius equals the product of the three sides divided by four times the area. The area comes from Heron's formula, so only the sides are needed.
Worked example
Find the circumscribed-circle radius of a triangle with sides 3, 4 and 5.
- Semiperimeter s = (3 + 4 + 5) / 2 = 6
- Area = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6
- R = (3 x 4 x 5) / (4 x 6) = 60 / 24 = 2.50
The circumscribed-circle radius is 2.50. These are the calculator's default inputs, so the result above matches the widget exactly.
Circumradius for common triangles
R = abc / (4 x Area).
| Sides | Area | Circumradius |
|---|---|---|
| 3, 4, 5 | 6.00 | 2.50 |
| 5, 12, 13 | 30.00 | 6.50 |
| 6, 8, 10 | 24.00 | 5.00 |
| 7, 24, 25 | 84.00 | 12.50 |
| 8, 15, 17 | 60.00 | 8.50 |
Mathematical reference: US National Institute of Standards and Technology (NIST).
Circumscribed Circle Radius Calculator: frequently asked questions
What is the circumscribed circle of a triangle?
It is the circle that passes through all three vertices of the triangle. Every triangle has exactly one such circle. Its center is the circumcenter, where the perpendicular bisectors of the sides meet, and its radius is the circumradius.
Where is the circumcenter located?
It depends on the triangle. For an acute triangle the circumcenter is inside, for a right triangle it lies exactly at the midpoint of the hypotenuse, and for an obtuse triangle it falls outside the triangle. The circumradius is measured from that center to any vertex.
What is the circumradius of a right triangle?
Exactly half the hypotenuse. Because the circumcenter sits at the hypotenuse midpoint, the radius equals half the longest side. For a 3, 4, 5 right triangle the hypotenuse is 5, so the circumradius is 2.5, which this formula confirms.
Do the three lengths have to form a valid triangle?
Yes. Each side must be less than the sum of the other two. If they do not satisfy that triangle inequality, no triangle exists, the Heron area becomes imaginary, and the circumradius is undefined. Check your inputs before relying on the result.
What is the circumradius formula?
The circumradius equals the product of the three sides divided by four times the area: R = abc / (4 x Area), with the area from Heron's formula. For a 3, 4, 5 triangle that is 60 divided by 24, which is 2.5.
Official sources
- Mathematical functions and computational reference data (Digital Library of Mathematical Functions): US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.