Inscribed Circle Radius Calculator
The inscribed circle of a triangle, or incircle, is the largest circle that fits entirely inside it, touching all three sides. Its radius, the inradius, is found from the triangle's area and its semiperimeter, which is half the sum of the three sides. This calculator takes the three side lengths and returns the inradius using the relationship that the inradius equals the area divided by the semiperimeter. The area itself is computed from the sides by Heron's formula, the square root of the semiperimeter times each semiperimeter-minus-side term, so you never need to know the height. The result is the radius of the circle that sits snugly inside the triangle, tangent to every side, centered at the incenter where the angle bisectors meet. Geometers, machinists, drafters, tilers and students use the inradius to fit round features inside triangular spaces and to characterize triangles. Enter the three sides to get the inradius immediately; the calculator also shows the area and semiperimeter it used. All three inputs are editable so you can test 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.
A triangle's inscribed-circle radius is area / semiperimeter. For sides 3, 4, 5 the area is 6 and the semiperimeter is 6, so the inradius is 1.00.
Inscribed Circle Radius 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 (incircle radius)
The inradius equals the triangle's area divided by its semiperimeter. The area is found from the sides by Heron's formula, so only the three sides are needed.
Worked example
Find the inscribed-circle radius of a triangle with sides 3, 4 and 5.
- Semiperimeter s = (3 + 4 + 5) / 2 = 6
- Area = sqrt(6 x (6-3)(6-4)(6-5)) = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6
- Inradius r = 6 / 6 = 1.00
The inscribed-circle radius is 1.00. 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 |
|---|---|---|
| 3, 4, 5 | 6.00 | 1.00 |
| 5, 12, 13 | 30.00 | 2.00 |
| 6, 8, 10 | 24.00 | 2.00 |
| 7, 24, 25 | 84.00 | 3.00 |
| 9, 12, 15 | 54.00 | 3.00 |
Mathematical reference: US National Institute of Standards and Technology (NIST).
Inscribed Circle Radius Calculator: frequently asked questions
What is the inscribed circle of a triangle?
It is the largest circle that fits inside the triangle and touches all three sides. Its center, the incenter, is the point where the three angle bisectors meet, and it is always inside the triangle. The radius of this circle is called the inradius.
How is the area found from the sides?
By Heron's formula. Compute the semiperimeter, half the sum of the three sides, then take the square root of the semiperimeter times each of the three semiperimeter-minus-side terms. This gives the area without needing any height or angle.
What makes three lengths a valid triangle?
Each side must be shorter than the sum of the other two, the triangle inequality. If that fails, the lengths cannot close into a triangle and the area under Heron's formula becomes imaginary. Check your inputs satisfy the inequality for a meaningful result.
Is the inradius the same as the apothem?
For a regular polygon the apothem is the inradius. For a general triangle the incircle radius found here is the equivalent inscribed-circle radius. Both describe the radius of the largest circle that fits inside the shape, tangent to its sides.
What is the inradius formula?
The inradius equals the triangle's area divided by its semiperimeter: r = Area / s, where s is half the sum of the sides. For a 3, 4, 5 triangle the area is 6 and the semiperimeter is 6, so the inradius is 1.
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.