Calculate the inscribed circle (incircle) of a triangle from three sides. Find inradius, incenter coordinates, incircle area, contact triangle, and circle-to-triangle area ratio.
The inscribed circle (or incircle) of a triangle is the largest circle that fits entirely inside the triangle. It is tangent to all three sides, and its center — the incenter — is the point where all three interior angle bisectors meet. The incenter is always located inside the triangle regardless of whether the triangle is acute, right, or obtuse.
The inradius r can be computed from the triangle's area A and semi-perimeter s using the elegant formula r = A / s. Once the three side lengths are known, Heron's formula supplies the area, which makes the incircle one of the cleanest ways to connect side data with an interior circle measurement.
This calculator accepts three side lengths, validates the triangle inequality, and reports the inradius, incircle area, tangent lengths, and area ratio so you can see how the circle fits inside the full triangle. That makes it useful both for textbook geometry and for design-style problems where internal clearances matter.
This calculator is useful when you need to understand how a triangle supports its largest interior circle, not just compute a single inradius value. It connects the area formula, the semi-perimeter, the tangent-length splits on each side, and the incircle coverage ratio in one place. That makes it practical for geometry study, drafting problems, packing or clearance questions, and any design where the largest circle inside a triangular boundary matters.
Semi-perimeter: s = (a + b + c) / 2 Triangle area (Heron's): A = √(s(s−a)(s−b)(s−c)) Inradius: r = A / s Incircle area: A_circle = πr² Tangent lengths: t_a = s − a, t_b = s − b, t_c = s − c Incenter: I = (a·A + b·B + c·C) / (a + b + c) Area ratio: R = πr² / A Contact triangle side: √((s−b)² + (s−c)² − 2(s−b)(s−c)·cos A)
Result: Inradius = 1, Incircle area ≈ 3.14, Triangle area = 6, Ratio ≈ 0.524
For the classic 3-4-5 right triangle: s = (3+4+5)/2 = 6, A = √(6·3·2·1) = 6, r = 6/6 = 1. Incircle area = π(1²) ≈ 3.14. The ratio 3.14/6 ≈ 0.524 means the incircle covers about 52.4% of the triangle.
The incircle is tied directly to the incenter, the point where the three internal angle bisectors meet. Because that point is always the same distance from all three sides, it becomes the only possible center of a circle tangent to every side. When you know that relationship, the inradius stops feeling like an isolated number and becomes part of the triangle's deeper center geometry.
Each side is split by the tangency point into two segments, and the equal tangents from the same vertex create the classic values s − a, s − b, and s − c. Those lengths are useful in proof problems and in understanding the contact triangle formed by the tangency points. Reporting them alongside the inradius helps you move from the global circle picture to the local side geometry.
The ratio of incircle area to triangle area tells you how efficiently the triangle contains a circle. Equilateral triangles make the best use of space, while flatter or more uneven triangles leave a larger fraction of the area outside the incircle. That ratio is a helpful way to compare shapes quickly, especially in packing, clearance, and optimization-style geometry questions.
The inscribed circle (incircle) is the largest circle that fits entirely inside the triangle, tangent to all three sides. Its center is the incenter, where the three angle bisectors meet.
Compute the semi-perimeter s = (a+b+c)/2 and area A via Heron's formula. Then r = A/s.
The contact (or intouch) triangle is formed by the three points where the incircle touches the sides. Its properties depend on the tangent lengths s−a, s−b, s−c.
No. The incircle is inside the triangle, tangent to all sides. The circumscribed circle (circumcircle) passes through all three vertices and is generally larger.
The equilateral triangle maximizes the ratio of incircle area to triangle area at π/(3√3) ≈ 0.6046 (about 60.5%).
You need at least one side length. With three angles and one side, use the law of sines to find the other sides, then compute the incircle normally.