Circumcircle Radius Calculator
Our free triangle calculator solves circumcircle radius problems. Get worked examples, visual aids, and downloadable results.
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.
How accurate are the results from Circumcircle Radius Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
Can I use Circumcircle Radius Calculator on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
What formula does Circumcircle Radius Calculator use?
The formula used is described in the Formula section on this page. It is based on widely accepted standards in the relevant field. If you need a specific reference or citation, the References section provides links to authoritative sources.