Circumscribed Circle Calculator — Circumradius
Find the circumscribed circle (circumcircle) — the unique circle that passes through all vertices — for a regular polygon or any triangle. Get the circumradius, circumference, and circle area.
Shape
Radius of the circle passing through all vertices
- 1
sin(π ÷ n)
sin(π ÷ 6) = 0.5 - 2
Circumradius R
5 ÷ (2 × 0.5) = 5From the isosceles triangle formed by the polygon's centre and two adjacent vertices.
How does this calculator work?
The circumscribed circle passes through all vertices of a polygon. For a regular n-gon with side a: R = a/(2 sin(π/n)). For a triangle with sides a, b, c: R = abc/(4·Area) where Area is from Heron's formula. Circumference of the circle = 2πR. Every triangle has exactly one circumcircle.
Formula
How this is calculated
The circumscribed circle (circumcircle) of a polygon is the unique circle that passes through every vertex. Every regular polygon and every triangle has exactly one such circle.
For a regular polygon with n sides of equal length a, the circumradius is R = a / (2 sin(π/n)). The formula comes from the isosceles triangle formed by the polygon's centre and any two adjacent vertices. The vertex angle at the centre is 2π/n, the two equal sides are both R, and the base (the polygon side) is a. From the base bisection: a/2 = R sin(π/n), giving R = a/(2 sin(π/n)). As n increases, sin(π/n) → π/n and R → an/(2π), which is the polygon's inscribed-radius plus a small correction.
For an arbitrary triangle with sides a, b, c and area S (computed by Heron's formula: S = √(s(s−a)(s−b)(s−c)) where s is the semi-perimeter), the circumradius is R = abc/(4S). This follows from the law of sines: a/sin A = 2R.
Frequently asked questions
A circumscribed circle (circumcircle) is the smallest circle that passes through all vertices of a polygon. Its centre is the circumcentre and its radius is the circumradius R. Every triangle and regular polygon has exactly one.
R = (6×6×6) / (4 × (√3/4 × 36)) = 216 / (4 × 9√3) = 6/√3 ≈ 3.464. Enter sides 6, 6, 6 in triangle mode to verify.
Yes. Every triangle has exactly one circumcircle. Its centre lies at the intersection of the perpendicular bisectors of the three sides, equidistant from all three vertices.
TG we-Calculate Editorial Team. (2026). Circumscribed Circle Calculator — Circumradius [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/circumscribed-circle-calculator
TG we-Calculate Editorial Team. "Circumscribed Circle Calculator — Circumradius." TG we-Calculate. 2026. https://we-calculate.com/calculator/circumscribed-circle-calculator.
TG we-Calculate Editorial Team, "Circumscribed Circle Calculator — Circumradius," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/circumscribed-circle-calculator
@misc{wecalculate_circumscribed_circle_calculator, title = {Circumscribed Circle Calculator — Circumradius}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/circumscribed-circle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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