Triangle Inradius Calculator
The inradius of a triangle is the radius of its inscribed circle, the largest circle that fits inside and touches all three sides. It is a compact measure of how much room a triangle has at its core, and it equals the triangle's area divided by its semiperimeter. This calculator takes the three side lengths and returns the inradius directly. It first finds the semiperimeter, half the sum of the sides, then the area by Heron's formula, the square root of the semiperimeter times each semiperimeter-minus-side term, and finally divides the area by the semiperimeter. Because everything follows from the three sides, you never need a height or an angle. The inradius is the distance from the incenter, where the three angle bisectors meet, to each side. Engineers, machinists, drafters and students use it to fit round features inside triangular frames and to compare triangles of different shapes. Enter the three sides to get the inradius immediately, with the area and semiperimeter shown so you can check the working. All three inputs are editable so you can model any valid triangle. Every figure here is computed deterministically from the formula shown below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
A triangle's inradius is area / s, where s is the semiperimeter. For sides 5, 12, 13 the area is 30 and s is 15, so the inradius is 2.00.
Triangle Inradius formula
r = Area / s
s = (a + b + c) / 2 (semiperimeter)
Area = sqrt( s (s-a)(s-b)(s-c) ) (Heron)
a, b, c = the three side lengths
r = inradius
The inradius is the triangle's area divided by its semiperimeter. The area is computed from the sides by Heron's formula, so only the three side lengths are required.
Worked example
Find the inradius of a triangle with sides 5, 12 and 13.
- Semiperimeter s = (5 + 12 + 13) / 2 = 15
- Area = sqrt(15 x (15-5)(15-12)(15-13)) = sqrt(15 x 10 x 3 x 2) = sqrt(900) = 30
- Inradius r = 30 / 15 = 2.00
The inradius is 2.00. These are the calculator's default inputs, so the result above matches the widget exactly.
Inradius for common triangles
r = Area / semiperimeter.
| Sides | Area | Inradius |
|---|---|---|
| 5, 12, 13 | 30.00 | 2.00 |
| 3, 4, 5 | 6.00 | 1.00 |
| 8, 15, 17 | 60.00 | 3.00 |
| 9, 40, 41 | 180.00 | 4.00 |
| 6, 8, 10 | 24.00 | 2.00 |
Mathematical reference: US National Institute of Standards and Technology (NIST).
Triangle Inradius Calculator: frequently asked questions
What is the inradius of a triangle?
It is the radius of the inscribed circle, the circle that fits inside the triangle touching all three sides. The inradius is the perpendicular distance from the incenter, where the angle bisectors meet, to each side. It measures the largest round feature that fits inside the triangle.
How does this differ from the apothem?
For a triangle the inradius is the apothem of that triangle, the center-to-side distance. The word apothem is more often used for regular polygons. Both describe the radius of the inscribed circle of the shape.
Why use Heron's formula here?
Heron's formula gives the area from the three side lengths alone, so you do not need a base and height. Since the inradius equals area divided by semiperimeter, computing the area by Heron lets the whole calculation run from just the sides.
What if the sides cannot form a triangle?
If any side is longer than the sum of the other two, the triangle inequality fails and no triangle exists. Heron's formula then yields an imaginary area and the inradius is undefined. The calculator only returns a meaningful result for valid triangles.
What is the inradius formula?
The inradius equals the triangle's area divided by its semiperimeter: r = Area / s. For a 5, 12, 13 triangle the area is 30 and the semiperimeter is 15, so the inradius is 30 divided by 15, which is 2.
Official sources
- Mathematical functions and computational reference data (Digital Library of Mathematical Functions): US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.