Triangle Circumradius Calculator
The circumradius of a triangle is the radius of its circumscribed circle, the unique circle that passes through all three vertices. It depends on the three side lengths and the triangle's area, and it equals the product of the three sides divided by four times the area. This calculator takes the three sides and returns the circumradius directly. It computes the area by Heron's formula, the square root of the semiperimeter times each semiperimeter-minus-side term, then divides the product of the sides by four times that area. Because everything follows from the sides, no angle or height is needed. The circumcircle is centered at the circumcenter, where the perpendicular bisectors of the sides meet, and the circumradius reaches from there to each vertex. For a right triangle the circumradius is exactly half the hypotenuse, a useful check. Surveyors, machinists, drafters and students use it when fitting a triangle inside a circular boundary or comparing triangle shapes. Enter the three sides to get the circumradius immediately, with the area shown so you can follow the working. Every figure here is computed deterministically from the formula shown below, with a worked example that reconciles exactly to the calculator.
A triangle's circumradius is abc / (4K), with K the area. For sides 5, 12, 13 the area is 30, so the circumradius is 6.50.
Triangle Circumradius formula
R = (a x b x c) / (4 K)
K = sqrt( s (s-a)(s-b)(s-c) ) (Heron area)
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 triangle's area K. The area is found from the sides by Heron's formula.
Worked example
Find the circumradius of a triangle with sides 5, 12 and 13.
- Semiperimeter s = (5 + 12 + 13) / 2 = 15
- Area K = sqrt(15 x 10 x 3 x 2) = sqrt(900) = 30
- R = (5 x 12 x 13) / (4 x 30) = 780 / 120 = 6.50
The circumradius is 6.50. These are the calculator's default inputs, so the result above matches the widget exactly.
Circumradius for common triangles
R = abc / (4K).
| Sides | Area | Circumradius |
|---|---|---|
| 5, 12, 13 | 30.00 | 6.50 |
| 3, 4, 5 | 6.00 | 2.50 |
| 8, 15, 17 | 60.00 | 8.50 |
| 7, 24, 25 | 84.00 | 12.50 |
| 6, 8, 10 | 24.00 | 5.00 |
Mathematical reference: US National Institute of Standards and Technology (NIST).
Triangle Circumradius Calculator: frequently asked questions
What is the circumradius of a triangle?
It is the radius of the circumscribed circle, the circle passing through all three vertices. Every triangle has exactly one circumcircle, centered at the circumcenter where the perpendicular bisectors of the sides meet. The circumradius is the distance from that center to any vertex.
Why divide by four times the area?
The relationship R = abc / (4K) comes from the law of sines combined with the area formula. It packages the side lengths and area into a single expression for the circumradius without needing any angle, which is why the area term appears in the denominator.
What is special about a right triangle?
For a right triangle the circumradius equals half the hypotenuse, because the circumcenter lies at the hypotenuse midpoint. A 5, 12, 13 triangle is right-angled, so its circumradius is half of 13, which is 6.5, matching the formula.
Do the sides need to form a valid triangle?
Yes. Each side must be shorter than the sum of the other two. If the triangle inequality is violated, no triangle exists, the Heron area is imaginary, and the circumradius is undefined. Use only side lengths that close into a triangle.
What is the circumradius formula?
The circumradius equals the product of the three sides divided by four times the area: R = abc / (4K). For a 5, 12, 13 triangle that is 780 divided by 120, which is 6.5.
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.