Calculate the circumscribed circle (circumcircle) of a triangle—the unique circle passing through all three vertices. Find radius, circumference, and area.
Last updated: April 2026 | By Patchworkr Team
A circumscribed circle (or circumcircle) is the unique circle that passes through all three vertices of a triangle. Every triangle has exactly one circumscribed circle.
Key properties:
For right triangles, the circumcenter is at the midpoint of the hypotenuse. For acute triangles, it's inside. For obtuse triangles, it's outside.
Check the triangle inequality: the sum of any two sides must exceed the third. For example, sides 3-4-5 work (3+4=7 > 5); sides 1-2-10 don't (1+2=3 < 10). Why: Only valid triangles have circumscribed circles. Invalid side combinations don't form a closed triangle and no circle can pass through three non-collinear points defined by invalid sides.
The semiperimeter is half the triangle's perimeter. Why: Heron's formula uses the semiperimeter to elegantly compute area without needing angles or heights. It's a clever algebraic tool that simplifies the mathematics.
Substitute the semiperimeter into Heron's formula. Why: Area is essential for the circumradius formula R = abc/(4A). Heron's formula lets you find area knowing only the three sides, without needing angles or height measurements.
Multiply all three side lengths, divide by four times the triangle area. Why: This formula is derived from the law of sines (a/sin(A) = 2R) combined with the area formula. It's the direct path from sides and area to the radius of the circle passing through all three vertices.
Once you have R, calculate circumference = 2πR and circle area = πR². Check: for right triangles, the radius should be half the hypotenuse. Why: Having the complete picture (radius, circumference, area) is useful for real applications like GPS networks, antenna placement, or geometric design. Verification catches computational errors before they propagate into real-world consequences.
GPS Triangulation
R = abc/(4A), where a, b, c are the triangle's side lengths and A is its area. Calculate area using Heron's formula first.
Yes! Every triangle has exactly one circumscribed circle. The center (circumcenter) is where the perpendicular bisectors of the sides meet.
For acute triangles, inside. For right triangles, at the hypotenuse midpoint. For obtuse triangles, outside the triangle.
A circumscribed circle passes through all vertices (outside the triangle). An inscribed circle fits inside, touching all three sides.
No. Only certain polygons have circumscribed circles (called cyclic polygons). All triangles are cyclic, but not all quadrilaterals are.
The law of sines states a/sin(A) = b/sin(B) = c/sin(C) = 2R, directly connecting side lengths, angles, and circumradius.
The sides must satisfy the triangle inequality: the sum of any two sides must be greater than the third. Otherwise, no triangle (or circle) exists.
For right triangles, the hypotenuse is a diameter of the circumscribed circle, so R = c/2 (where c is the hypotenuse). The circumcenter is at the hypotenuse's midpoint.