What this calculator does
This tool finds the incircle (inscribed circle) of any triangle from its three side lengths. The incircle is the largest circle that fits inside a triangle, tangent to all three sides, and its center is the incenter. The calculator returns the incircle radius (r), diameter, the incircle area, the triangle area, and the ratio of the incircle area to the triangle area. All sides must be entered in the same unit "L"; radius and diameter come out in L and areas in L², with no unit conversion applied.
How to use it
Enter the three side lengths a, b and c. They must all be positive and satisfy the triangle inequality — each side must be strictly smaller than the sum of the other two. If they do not form a valid triangle, the calculator reports an error instead of a meaningless result.
The formula explained
First compute the semiperimeter \(s = (a + b + c) / 2\). Heron's formula gives the triangle area $$S_t = \sqrt{s(s-a)(s-b)(s-c)}$$ The incircle radius is simply the area divided by the semiperimeter: $$r = \frac{S_t}{s}$$ From there the diameter is \(\phi = 2r\), the incircle area is \(S_c = \pi r^{2}\), and the area ratio is \(S_c / S_t\).
Worked example
For the classic 3-4-5 right triangle: $$s = \frac{3+4+5}{2} = 6$$ $$S_t = \sqrt{6\cdot 3\cdot 2\cdot 1} = \sqrt{36} = 6$$ So \(r = 6/6 = 1\), diameter = 2, incircle area = \(\pi\cdot 1^{2} \approx 3.14159\), and the area ratio = \(3.14159 / 6 \approx 0.5236\).
FAQ
What is the incircle? It is the unique circle inside a triangle that touches all three sides. Its center is the incenter, where the angle bisectors meet.
Why do I get an error? Your sides do not form a real triangle — either a side is zero/negative or one side is longer than the other two combined.
Can it handle any triangle? Yes, scalene, isosceles, equilateral and right triangles all work, as long as the triangle inequality holds.
