This calculator finds the circumscribed circle of any triangle, the single circle that passes through all three vertices, and returns its radius, circumference and area without you working through the trigonometry yourself. Enter the three side lengths into the Side a, Side b and Side c fields and choose a decimal place setting; the tool checks the lengths form a valid triangle, then applies the standard circumradius formula R = abc / (4A), where A is the triangle's area from Heron's formula. Results shown are the circumradius, diameter, circumference and area of the circumcircle, plus a calculation breakdown covering the semi-perimeter, triangle area, the abc product and the 4A term, so you can follow every step. It also detects and displays the triangle type, right-angled, acute, obtuse, equilateral or isosceles, since this affects where the circumcentre sits. Use it to check geometry homework, verify circular layouts built around a triangular frame, or see how a triangle's shape determines its circumcircle. For a right-angled triangle the circumradius always equals half the hypotenuse, a useful way to sanity check your inputs. The formula is exact for any valid triangle, so results are precise rather than approximate, though they are only as accurate as the side lengths you enter.
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Standard formula
Circumradius formula R = abc / (4A) via Heron's area. Exact for any valid triangle.
Triangle Side Lengths
units
units
units
Formula Used
R = (a × b × c) / (4 × A)
where A is the triangle area from Heron's formula:
s = (a + b + c) / 2
A = √(s(s−a)(s−b)(s−c))
Right-angled (3-4-5)
These side lengths do not form a valid triangle. Each side must be shorter than the sum of the other two sides.
Circumscribed Circle Results
Circumradius (R)
2.500
units
Diameter (2R)
5.000
units
Circumference
15.708
units (2πR)
Circumcircle Area
19.635
square units (πR²)
Calculation Breakdown
Side a3
Side b4
Side c5
Semi-perimeter (s)6.000
Triangle area (A)6.000
a × b × c60
4 × A24.000
Circumradius R2.500
Worked Example (3-4-5 Triangle)
s = (3+4+5)/26
A = √(6×3×2×1)6
abc = 3×4×560
4A = 4×624
R = 60/242.5
Circumference = 2π×2.515.708
Circle area = π×2.5²19.635
Result:A 3-4-5 right-angled triangle has a circumradius of 2.500 units. The circumscribed circle has a circumference of 15.708 units and an area of 19.635 square units.
What Is a Circumscribed Circle?
The circumscribed circle (circumcircle) of a triangle is the unique circle that passes through all three vertices of the triangle. Its centre, the circumcentre, is equidistant from all three vertices. Every triangle, regardless of its shape, has exactly one circumscribed circle.
The distance from the circumcentre to any vertex is the circumradius, denoted R. The circumradius is the defining measure of the circumcircle: once you know R, you can calculate the circumference (2πR) and area (πR²) of the circle directly.
The Circumradius Formula
For a triangle with sides a, b, c and area A, the circumradius is:
R = (a × b × c) / (4 × A)
The area A is found from Heron's formula. First calculate the semi-perimeter s = (a + b + c) / 2, then:
A = √(s × (s − a) × (s − b) × (s − c))
Substituting back, you get a formula entirely in terms of the side lengths. This formula works for any valid triangle: acute, obtuse, or right-angled.
Special Cases
Triangle type
Circumradius property
Right-angled
R = hypotenuse / 2 (circumcentre is at the midpoint of the hypotenuse)
Equilateral (side a)
R = a / √3 (circumcentre is at the centroid)
Acute triangle
Circumcentre lies inside the triangle
Obtuse triangle
Circumcentre lies outside the triangle
Circumcircle vs Incircle
The circumcircle passes through all three vertices of the triangle. The incircle (inscribed circle) is tangent to all three sides and fits inside the triangle. These are different circles with different radii. The inradius r = A / s, where s is the semi-perimeter. The relationship R × r = abc / (4s) connects the two.
Worked Example
For a triangle with sides a = 3, b = 4, c = 5 (a 3-4-5 right triangle):
Method: Circumradius formula R = abc / (4A) with Heron's formula for triangle area. See: Weisstein, Eric W., "Circumradius", MathWorld (mathworld.wolfram.com/Circumradius.html).
This calculator works for any valid triangle (where each side is shorter than the sum of the other two). Results are given to the selected number of decimal places. The formula is exact for any triangle that satisfies the triangle inequality.
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