GeomBench

Circumradius Calculator

The circle that touches every corner.

Side a

Side b

Side c

Circumradius

3.572

What you entered

Sides: 5, 6, 7

1
Area (Heron's formula)
√(9 × 4 × 3 × 2)= 14.6969
2
Circumradius R = (a × b × c) ÷ (4 × Area)
(5 × 6 × 7) ÷ (4 × 14.6969)= 3.5722

Result

Circumradius: 3.572

The circumradius (radius of the circle passing through every vertex) is 3.572 units.

Circumradius of triangles and regular polygons

The circumradius is the radius of the circle passing through every vertex. Inradius is shown alongside.

Shape	Dimensions	Circumradius	Inradius
Triangle	3, 4, 5	2.5000	1.0000
Triangle	6, 8, 10	5.0000	2.0000
Triangle	5, 12, 13	6.5000	2.0000
Polygon	6 sides, side 4	4.0000	3.4641
Polygon	5 sides, side 6	5.1039	4.1291

The hexagon row is a special case worth knowing: a regular hexagon's circumradius exactly equals its side length, which is why six equilateral triangles tile perfectly into one and why hexagonal packing is so efficient. For triangles the circumradius follows from abc divided by four times the area, and in every right triangle here it lands on half the hypotenuse. The circumradius is always at least twice the inradius, with equality only for an equilateral triangle - a result known as Euler's inequality.

Every triangle has exactly one circumscribed circle

R = (a×b×c) ÷ (4×Area) gives the radius of the unique circle passing through all three vertices of any triangle — useful in trigonometry (it's also the constant in the Law of Sines: a/sin(A) = 2R).

Regular polygons are simpler

For a regular polygon, every vertex is automatically equidistant from the center, so the circumradius has a direct formula (s ÷ (2·sin(π/n))) rather than needing Heron's formula first.

Frequently asked questions

What is the circumradius of a triangle?

The circumradius is the radius of the circle passing through all three vertices. Formula: R = (a×b×c) / (4×Area). For a 3-4-5 triangle: R = (3×4×5) / (4×6) = 60/24 = 2.5. The circumcircle is unique for every triangle.

How are circumradius and inradius related?

For any triangle, the circumradius R is always ≥ 2 times the inradius r (Euler's inequality: R ≥ 2r). Equality holds only for equilateral triangles. For an equilateral triangle with side a: R = a/√3, r = a/(2√3), so R = 2r.

What is the circumradius of a regular polygon?

R = s / (2×sin(π/n)), where s is the side length and n is the number of sides. For a regular hexagon with side 10: R = 10 / (2×sin(30°)) = 10 / 1 = 10. A regular hexagon's circumradius equals its side length.