Circumcircle Radius Calculator
Our free triangle calculator solves circumcircle radius problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
R = (a x b x c) / (4 x Area)
Where R is the circumradius, a, b, c are the three sides of the triangle, and Area is calculated using Herons formula: Area = sqrt(s(s-a)(s-b)(s-c)) with s = (a+b+c)/2. Equivalently, R = a / (2 sin A) from the law of sines.
Worked Examples
Example 1: Circumradius from Three Sides
Problem:Find the circumradius of a triangle with sides 7, 8, and 9.
Solution:Semi-perimeter s = (7 + 8 + 9) / 2 = 12\nArea = sqrt(12 x (12-7) x (12-8) x (12-9)) = sqrt(12 x 5 x 4 x 3) = sqrt(720) = 26.8328\nCircumradius R = (7 x 8 x 9) / (4 x 26.8328) = 504 / 107.3313 = 4.6953\nCircumscribed circle area = pi x 4.6953^2 = 69.26 sq units
Result:Circumradius = 4.6953 | Circle Area = 69.26 sq units | Triangle Area = 26.8328 sq units
Example 2: Circumradius of a Right Triangle
Problem:Find the circumradius of a right triangle with legs 5 and 12.
Solution:Hypotenuse = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13\nCircumradius R = hypotenuse / 2 = 13 / 2 = 6.5\nTriangle area = (1/2) x 5 x 12 = 30\nVerify: R = (5 x 12 x 13) / (4 x 30) = 780 / 120 = 6.5
Result:Circumradius = 6.5 | Hypotenuse = 13 | Triangle Area = 30 sq units
Frequently Asked Questions
What is a circumcircle and what is its radius?
A circumcircle (also called a circumscribed circle) is the unique circle that passes through all three vertices of a triangle. Every triangle has exactly one circumcircle, and its center is called the circumcenter. The circumradius is the radius of this circle, which equals the distance from the circumcenter to any vertex. The circumcenter is found at the intersection of the perpendicular bisectors of the three sides. For acute triangles, the circumcenter lies inside the triangle; for right triangles, it is the midpoint of the hypotenuse; for obtuse triangles, it lies outside the triangle.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy