Circumcircle Radius Calculator
Our free triangle calculator solves circumcircle radius problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Circumcircle Radius Calculator
Calculator
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Formula: R = (a x b x c) / (4 x Area)
Worked example — Circumradius = 4.6953 | Circle Area = 69.26 sq units | Triangle Area = 26.8328 sq units
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 Area = sqrt(12 x (12-7) x (12-8) x (12-9)) = sqrt(12 x 5 x 4 x 3) = sqrt(720) = 26.8328 Circumradius R = (7 x 8 x 9) / (4 x 26.8328) = 504 / 107.3313 = 4.6953 Circumscribed 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 Circumradius R = hypotenuse / 2 = 13 / 2 = 6.5 Triangle area = (1/2) x 5 x 12 = 30 Verify: 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 do you find the circumradius of a right triangle?
For a right triangle, the circumradius has a beautifully simple formula: R = hypotenuse / 2. This is because the hypotenuse is always the diameter of the circumscribed circle in a right triangle. The circumcenter is located exactly at the midpoint of the hypotenuse. This property follows from Thales theorem, which states that any angle inscribed in a semicircle is a right angle. So if the right angle vertex lies on the circle and subtends the hypotenuse, the hypotenuse must be the diameter. This makes right triangles the easiest case for circumradius calculations.
What is the circumradius of an equilateral triangle?
For an equilateral triangle with side length s, the circumradius equals s / sqrt(3), which can also be written as s times sqrt(3) / 3. This is exactly twice the inradius, which equals s times sqrt(3) / 6. The circumcenter of an equilateral triangle coincides with the centroid, incenter, and orthocenter, since all four triangle centers merge into a single point for equilateral triangles. The circumscribed circle area is (pi times s squared) / 3. For example, an equilateral triangle with side 6 has circumradius = 6 / sqrt(3) = 3.464 and inradius = 1.732.
How does the circumradius relate to the law of sines?
The law of sines directly involves the circumradius: a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius. This means any side divided by the sine of its opposite angle equals twice the circumradius. This relationship provides an alternative way to compute R when you know one side and its opposite angle: R = a / (2 sin(A)). The law of sines is one of the most fundamental theorems in trigonometry and geometry, connecting side lengths, angles, and the circumscribed circle into a single elegant equation.
Can the circumcenter be outside the triangle?
Yes, the circumcenter lies outside the triangle when the triangle is obtuse (has an angle greater than 90 degrees). Specifically, the circumcenter is on the opposite side of the longest edge from the obtuse angle vertex. For an acute triangle, the circumcenter is inside the triangle. For a right triangle, the circumcenter is exactly at the midpoint of the hypotenuse, which is on the boundary of the triangle. This behavior contrasts with the centroid, which always lies inside the triangle. The position of the circumcenter relative to the triangle is determined entirely by the largest angle.
What are the practical applications of the circumradius?
The circumradius has numerous practical applications across multiple fields. In engineering, it helps determine the minimum bounding circle for triangular components, which is critical for packaging and machining operations. In computer graphics, circumscribed circles are used in Delaunay triangulation for mesh generation and terrain modeling. In navigation and surveying, the circumradius helps solve triangulation problems for position determination. Astronomers use it in orbital mechanics calculations. Architects apply circumradius calculations when designing triangular structures that must fit within circular constraints, such as decorative windows.
How is the circumradius used in Euler line calculations?
The circumcenter is one of the key points on the Euler line, which also passes through the centroid and orthocenter of a triangle. The circumcenter C, centroid G, and orthocenter H are collinear (lie on the same line), and the centroid divides the segment from circumcenter to orthocenter in a 1:2 ratio (CG:GH = 1:2). The nine-point circle, whose center also lies on the Euler line, has a radius equal to exactly half the circumradius. These relationships connect the circumradius to many other important triangle measurements and demonstrate the deep geometric structure underlying triangle geometry.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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