Circumradius and Inradius Calculator
Enter the three side lengths of a triangle to find its circumradius, inradius, area, and semi-perimeter.
units
units
units
Radius of the circumscribed circle
- 1
Semi-perimeter
(6 + 8 + 10) ÷ 2 = 12 - 2
Area (Heron)
√(12 × 6 × 4 × 2) = 24 - 3
Circumradius R = a × b × c ÷ (4 × Area)
6 × 8 × 10 ÷ (4 × 24) = 5R is the radius of the unique circle passing through all three vertices.
How does this calculator work?
From three side lengths a, b, c, compute the semi-perimeter s = (a+b+c)/2 and Heron area √(s(s−a)(s−b)(s−c)). The circumradius is R = abc/(4·Area) and the inradius is r = Area/s. Sides must satisfy the triangle inequality; R is always at least twice r.
Formula
How this is calculated
The three inputs a, b, and c are the side lengths of a triangle, measured in any consistent unit of length. The sides must be positive and satisfy the triangle inequality — each side must be shorter than the sum of the other two — otherwise no triangle exists and the result is undefined.
The semi-perimeter s is half the perimeter, s = (a + b + c) / 2. Heron's formula gives the area as √(s(s−a)(s−b)(s−c)). The circumradius (radius of the unique circle passing through all three vertices) is R = abc / (4·Area), and the inradius (radius of the largest circle that fits inside the triangle, tangent to all three sides) is r = Area / s.
The area is reported in squared length units, while R and r share the same length unit as the inputs. As a triangle becomes degenerate (the three points nearly collinear) the area approaches zero, the inradius shrinks to zero, and the circumradius grows without bound. Euler's inequality guarantees R ≥ 2r for any valid triangle, with equality only for an equilateral triangle.
Frequently asked questions
The circumradius R is the radius of the circle that passes through all three vertices (the circumscribed circle). The inradius r is the radius of the circle inscribed inside the triangle, tangent to all three sides.
The sides must all be positive and obey the triangle inequality: every side must be less than the sum of the other two. If they do not, no triangle can be formed and the radii are undefined.
Yes. Euler's inequality states R ≥ 2r for every triangle, with equality reached only by the equilateral triangle.
Also known as
TG we-Calculate Editorial Team. (2026). Circumradius and Inradius Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/triangle-circumradius-inradius-calculator
TG we-Calculate Editorial Team. "Circumradius and Inradius Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/triangle-circumradius-inradius-calculator.
TG we-Calculate Editorial Team, "Circumradius and Inradius Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/triangle-circumradius-inradius-calculator
@misc{wecalculate_triangle_circumradius_inradius_calculator, title = {Circumradius and Inradius Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/triangle-circumradius-inradius-calculator}}, year = {2026}, note = {TG we-Calculate} }
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