How to Calculate Inscribed Circle
What is Inscribed Circle?
The inscribed circle (incircle) is the largest circle that fits inside a triangle, tangent to all three sides. Its radius (inradius) equals the triangle area divided by the semi-perimeter.
Formula
r = A/s where A = area, s = semi-perimeter
- r
- inradius (inscribed circle radius) (length)
- A
- triangle area (length²)
- s
- semi-perimeter (length)
Step-by-Step Guide
- 1r = Area / s
- 2Where s = (a+b+c)/2 (semi-perimeter)
- 3Area found via Heron's formula
- 4Centre is at the intersection of the angle bisectors (incentre)
Worked Examples
Input
Triangle 3, 4, 5
Result
r = 6/6 = 1
Input
Triangle 5, 12, 13
Result
r = 30/15 = 2
Frequently Asked Questions
What is the incentre of a triangle?
The incentre is the center of the inscribed circle, located at the intersection of the three angle bisectors.
Is the inradius always smaller than the circumradius?
Yes, for any triangle, the inradius r is always less than the circumradius R.
How many inscribed circles can a triangle have?
Exactly one. The inscribed circle is unique and is the largest circle that fits inside the triangle.
Ready to calculate? Try the free Inscribed Circle Calculator
Try it yourself →