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Triangle Circumradius Calculator

Find the circumradius of a triangle.

How it is calculated

Enter the three side lengths in any one unit. The tool first computes the semi-perimeter s = (a + b + c) ÷ 2, then the area with Heron's formula Area = √(s(s − a)(s − b)(s − c)), and finally R = a × b × c ÷ (4 × Area). R is shown with four decimals in the same length unit as the sides, and the area with two decimals in squared units (the output is labelled generic 'units'). All sides must be above zero and must satisfy the triangle inequality: each pair of sides must add up to strictly more than the third. If one side equals the sum of the other two the triangle is flat, and the tool returns no result.

Worked example

Take a triangle with sides 13, 14 and 15. The semi-perimeter is (13 + 14 + 15) ÷ 2 = 21. Heron's formula gives √(21 × 8 × 7 × 6) = √7056 = 84, so the area is 84.00. The product of the sides is 13 × 14 × 15 = 2730. Then R = 2730 ÷ (4 × 84) = 2730 ÷ 336 = 8.1250 units. A quick check: the circumradius must be at least half the longest side, 15 ÷ 2 = 7.5, and 8.125 satisfies that.

Try this example →

Common mistakes

  • Entering sides that do not form a triangle, such as 3, 4 and 7, where 3 + 4 equals 7; the tool returns no result for these.
  • Mixing units between sides, for example one side in metres and another in centimetres.
  • Confusing the circumradius with the inradius; the circumscribed circle passes through the three vertices and is always at least as large as the inscribed circle.
  • Expecting the circumcentre to lie inside the triangle; in an obtuse triangle it lies outside, though R is still correct.
  • Entering sides rounded too coarsely for a thin triangle; when the triangle is nearly flat the area is tiny and R changes a lot with small errors in the sides.

About this calculator

The circumradius of a triangle — the radius of the unique circle passing through all three vertices — can be found from the three side lengths as R = abc/(4·Area), where Area comes from Heron's formula, or equivalently from the law of sines as R = a/(2·sin A) using any side and its opposite angle. Every triangle has exactly one circumscribed circle, whose centre (the circumcenter) is the intersection of the perpendicular bisectors of the sides.

This relation is central to trigonometry coursework and to any application that needs a circle fitted exactly through three points: surveyors and civil engineers use it to lay out curved roads and arcs from three known points, machinists and CNC programmers use it to compute bolt-circle or fillet radii, and computational-geometry and computer-graphics code uses it for triangle mesh analysis and Delaunay triangulation.

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Comments (1)

  • Editorial team

    Two shape checks help. For a right triangle the circumradius is exactly half the hypotenuse, so sides 3, 4 and 5 give 2.5. For an obtuse triangle the circle's centre lies outside the triangle, and a very flat triangle gives an enormous radius because its area shrinks toward zero. Are you working from three sides, or do you have angles that you would like supported?

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