How to use this calculator
- Enter or choose i know.
- Enter or choose length unit.
- The inradius r updates as you type. Use Copy result or Share to keep it.
Method
r = A ÷ s; s = (a + b + c) ÷ 2; A = √(s(s − a)(s − b)(s − c)) (Heron)
- r
- inradius
- A
- triangle area
- s
- semiperimeter
- a, b, c
- side lengths
Worked example
- s = (3 + 4 + 5) ÷ 2 = 6.
- Area = √(6 × 3 × 2 × 1) = 6 cm².
- r = 6 ÷ 6 = 1 cm.
The 3-4-5 right triangle has an inscribed circle of radius exactly 1 cm.
What the result means
The incircle’s centre (the incentre) is where the three angle bisectors meet. The inradius is the largest round piece you can cut from a triangular sheet.
Assumptions and limits
- The sides form a valid, non-degenerate triangle.
- In area-and-perimeter mode the tool only checks that the area is possible for the perimeter (at most P² ÷ (12√3)); it cannot tell which triangle you have.
- The circumscribed circle (circumradius) is a different measure, R = abc ÷ 4A.
Questions
Why does r = A ÷ s work?
Lines from the incentre to the corners cut the triangle into three triangles of height r on bases a, b and c, so A = ½r(a + b + c) = r·s.
Is there a shortcut for right triangles?
Yes: r = (a + b − c) ÷ 2, where c is the hypotenuse. For 3-4-5 that gives (3 + 4 − 5) ÷ 2 = 1.