Triangle Inradius Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The inradius of a triangle (the largest circle that fits inside, tangent to all three sides) is calculated with r = Area ÷ s, where s is the semi-perimeter and the area is obtained by Heron's formula. For a right triangle with sides 3, 4, and 5, this radius is exactly 1.
Explanation
The incircle of a triangle is the largest circle that fits entirely inside the figure, tangent to all three sides at once; its center, called the incenter, is the intersection point of the three angle bisectors of the triangle. Its radius is calculated with r = Area ÷ s, where s = (a+b+c)/2 is the semi-perimeter and the area comes from our triangle area calculator (Heron's formula), reused directly internally by this calculator. This inradius complements the site's circumradius calculator: the two notable circles of a triangle, one contained inside and tangent to the sides, the other passing through the three vertices and containing it entirely, are calculated from the same three side lengths but with different formulas (r = Area/s versus R = abc/(4×Area)). A special case allows this calculation to be checked independently: for any right triangle with hypotenuse c and legs a and b, a classic geometric identity gives r = (a + b − c) ÷ 2 — applied to the 3-4-5 triangle, this gives (3+4−5)/2 = 1, exactly the value obtained by the general formula. The inradius comes up in architecture and design to inscribe the largest possible circle in a triangular template, as well as in geometric optimization.
Example: right triangle with sides 3, 4, and 5
Inputs
Sides: a = 3 m, b = 4 m, c = 5 m.
Calculation
s = (3+4+5) ÷ 2 = 6. Area = √(6×3×2×1) = √36 = 6 m². r = 6 ÷ 6 = 1 m.
Result
The inradius of this right triangle is 1 m, confirmed independently by (3+4−5)÷2 = 1.
Frequently asked questions
What is the difference between the incircle and the circumcircle?
The incircle is the largest circle contained inside the triangle, tangent to its three sides (radius r = Area/s). The circumcircle is instead the circle passing exactly through the three vertices and containing the whole triangle (radius R = abc/(4×Area)). For a given triangle, these two radii are almost always different: R is always at least twice as large as r (equality only for an equilateral triangle).
What is the incenter of a triangle?
The incenter is the center of the incircle: it is the point where the three angle bisectors of the triangle cross. Unlike the center of the circumcircle, which can lie outside an obtuse triangle, the incenter is always located inside the triangle, whatever its shape.
Why does this calculator show an empty result for some values?
Three lengths do not always form a valid triangle: they must satisfy the triangle inequality (the sum of two sides must always exceed the third). If not, as for sides 1, 1, and 10, no real triangle exists and neither the area nor the inradius can be calculated.