Triangle Circumradius Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/5/2026
A triangle's circumscribed circle radius (the one circle that passes through all three vertices) is calculated with R = (a × b × c) ÷ (4 × Area), the area obtained via Heron's formula. For a right triangle with sides 3, 4, and 5, this radius is exactly 2.5 — half the hypotenuse.
Explanation
A triangle's circumscribed circle is the one circle that passes exactly through its three vertices; its radius is calculated by combining the three side lengths with the triangle's area, itself obtained the same way as in our triangle area calculator (base × height ÷ 2 for a simple case, or Heron's formula from three sides in the general case used internally here). This calculator applies R = (a×b×c) ÷ (4×Area) once that area is known. One notable special case lets this formula be checked in a completely independent way: for any right triangle, a classic geometry theorem (the inscribed angle theorem, applied to a right angle) establishes that the circumradius equals exactly half the hypotenuse — confirmed by our two test cases on the 3-4-5 and 5-12-13 right triangles (R = 5÷2 = 2.5 and R = 13÷2 = 6.5), calculated here by the general formula but landing exactly on this independently known value; see our Pythagorean hypotenuse calculator for that same relationship applied directly. The circumradius comes up in many practical contexts: in architecture and design to fit a triangle inside a given circular shape, or in trigonometry, where it appears directly in a generalized version of the law of sines (a ÷ sin(A) = 2R, relating each side to its opposite angle and the circumradius).
Example: a right triangle with sides 3, 4, and 5
Inputs
Sides: a = 3 m, b = 4 m, c = 5 m.
Calculation
s = (3+4+5) ÷ 2 = 6. Area = √(6×3×2×1) = √36 = 6 m². R = (3×4×5) ÷ (4×6) = 60 ÷ 24 = 2.5 m.
Result
This right triangle's circumradius is 2.5 m, exactly half its hypotenuse (5 m).
Frequently asked questions
Why does a right triangle's circumradius always equal half its hypotenuse?
This follows from the inscribed angle theorem: in a circle, an inscribed angle that intercepts a diameter is always exactly 90°. Conversely, if a triangle has a right angle, its hypotenuse must be a diameter of the circumscribed circle — so the radius equals half that diameter, meaning half the hypotenuse, regardless of the other two sides.
What's the circumradius formula for an equilateral triangle?
For an equilateral triangle with side a, the circumradius simplifies to R = a ÷ √3 (about 0.577 × a), a formula that follows directly from the general case applied to three equal sides. A triangle with side 6, for example, has a circumradius of about 3.46, as confirmed by this calculator's third test case.
Does this formula work for any triangle?
Yes, the formula R = (a×b×c) ÷ (4×Area) holds for any valid triangle, whether right, isosceles, equilateral, or scalene — the only condition being that the three lengths must actually be able to form a triangle (satisfying the triangle inequality), otherwise the area given by Heron's formula doesn't really exist.