Why does r equal area divided by the semi-perimeter?

Joining the incenter to each vertex cuts the triangle into three triangles of height r, whose areas add to ½r(a + b + c) = r·s.

Is there a shortcut for right triangles?

Yes: r = (a + b − c) / 2. For 3, 4 and 5 that gives (3 + 4 − 5) / 2 = 1, the same as area ÷ semi-perimeter.

How does the incenter differ from the circumcenter?

The incenter is equidistant from the sides and always inside the triangle; the circumcenter is equidistant from the vertices and can fall outside an obtuse triangle.

⭕ Inscribed Circle Calculator

Calculate the inscribed circle radius and area of a triangle.

Side a Side b Side c

What is the Inscribed Circle?

The circle tangent to all three sides of a triangle. Its center (incenter, I) is the intersection of the three angle bisectors.

Inscribed Circle Radius Formula

r = S / s

S is the area of the triangle and s is the semi-perimeter (perimeter / 2).

s = (a + b + c) / 2

S = √(s(s−a)(s−b)(s−c))

Alternative Expression

r = (a + b − c) / 2 × tan(C/2)

This formula shows that for a right triangle (C=90°), r = (a + b − c) / 2.

Relation to Area

S = r × s

The product of the inradius and the semi-perimeter equals the triangle area.

Incenter Coordinates

I = (a·A + b·B + c·C) / (a + b + c)

The incenter is the weighted average of the vertices, with each side length as the weight.

Worked examples

Sides 3, 4 and 5 have a semi-perimeter of 6 and an area of 6, so r = 1, the circle area is π ≈ 3.1416 and the circumference about 6.2832.

An equilateral triangle of side 6 gives r = 15.5885 ÷ 9 ≈ 1.7321, a circle area of 3π ≈ 9.4248 and a circumference of about 10.8828.

What the result assumes

The circle touches each side exactly once, at the foot of the perpendicular from the incenter, which is where the three angle bisectors meet. It is the largest circle that fits inside the triangle, and its radius stays below the smallest altitude. Impossible side triples are rejected before any radius is computed.

Frequently asked questions

Why does r equal area divided by the semi-perimeter?: Joining the incenter to each vertex cuts the triangle into three triangles of height r, whose areas add to ½r(a + b + c) = r·s.
Is there a shortcut for right triangles?: Yes: r = (a + b − c) / 2. For 3, 4 and 5 that gives (3 + 4 − 5) / 2 = 1, the same as area ÷ semi-perimeter.
How does the incenter differ from the circumcenter?: The incenter is equidistant from the sides and always inside the triangle; the circumcenter is equidistant from the vertices and can fall outside an obtuse triangle.