Circumscribed Circle Calculator

Enter the three side lengths of a triangle to find the radius of its circumscribed circle, along with the triangle's area and the circle's area and circumference.

Quick Facts

Formula
R = (a × b × c) / (4 × Area)
Area comes from Heron's formula: s = (a+b+c)/2, Area = sqrt(s(s−a)(s−b)(s−c)).
Property
The circumcenter is equidistant from all three vertices
It sits inside an acute triangle, on the hypotenuse of a right triangle, and outside an obtuse triangle.

Your Results

Calculated
Circumradius (R)
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Radius of the circumscribed circle
Triangle area
-
Via Heron's formula
Circle circumference
-
2 × π × R
Circle area
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π × R²

Ready

Enter the three side lengths of a triangle, then press Calculate.

Formula and method for the circumscribed circle

The circumscribed circle (or circumcircle) of a triangle is the unique circle that passes through all three of its vertices. Its center, the circumcenter, is the point equidistant from all three corners — the intersection of the triangle's three perpendicular bisectors. This calculator finds that circle's radius directly from the triangle's three side lengths.

How the calculation works

Given side lengths a, b, and c, the calculator first finds the triangle's area with Heron's formula: compute the semi-perimeter s = (a + b + c) / 2, then Area = √(s(s−a)(s−b)(s−c)). The circumradius then follows from R = (a × b × c) / (4 × Area). This is exact for any valid triangle — no angles need to be entered or measured. From R, the circle's circumference is 2πR and its area is πR². Enter all three sides in the same unit; the results come out in that unit (and its square, for areas).

Common mistakes

  • Invalid triangles: three lengths only form a triangle if each side is shorter than the sum of the other two (the triangle inequality). 2, 3, and 10 cannot form a triangle, so no circumscribed circle exists for them.
  • Mixed units: entering one side in inches and another in centimeters silently corrupts the result. Convert every side to the same unit before entering it.
  • Confusing circumradius with inradius: the circumcircle passes through the vertices and sits outside (or on) the triangle's edges; the inscribed circle instead touches each side and sits entirely inside. They use different formulas and are rarely the same size.

Real-world applications

  • Manufacturing and CNC/laser cutting use the circumradius to find the smallest circular stock or bounding disk that a triangular part will fit inside.
  • Structural and truss design uses the circumcenter and circumradius to check clearances and to place circular fittings or pins that must touch three structural points.
  • Surveying and navigation use the same three-point-to-circle relationship (trilateration) to locate a point equidistant from three known reference points.
  • Geometry and trigonometry coursework uses the circumradius formula alongside the Law of Sines, since R also equals a / (2 sin A) for any angle A opposite side a.

Frequently Asked Questions

What is the formula for the circumradius of a triangle?
For a triangle with sides a, b, and c, the circumradius is R = (a × b × c) / (4 × Area), where Area comes from Heron's formula: s = (a+b+c)/2 and Area = √(s(s−a)(s−b)(s−c)). The circumscribed circle passes through all three vertices of the triangle.
Can any three side lengths form a triangle?
No. Three lengths only form a valid triangle if each side is shorter than the sum of the other two (the triangle inequality: a+b>c, b+c>a, and a+c>b). If any of those fail, the calculator flags the input as invalid because no such triangle, and therefore no circumscribed circle, exists.
Where is the center of the circumscribed circle located?
The circumcenter sits inside the triangle when all angles are acute, exactly on the midpoint of the hypotenuse when the triangle is right-angled, and outside the triangle when one angle is obtuse. The calculator classifies your triangle as acute, right, or obtuse based on this same relationship.
What is the circumradius of a right triangle?
For a right triangle, the hypotenuse is a diameter of the circumscribed circle, so the circumradius is simply half the hypotenuse: R = c/2. For example, a 3-4-5 right triangle has a circumradius of 5/2 = 2.5.

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