Every triangle has two special circles. The circumcircle passes through all three vertices; the incircle sits inside, touching all three sides. Their radii and centers come up in construction, surveying, mesh generation and countless geometry problems. This calculator finds both from either three side lengths or three vertex coordinates, checks the result with Euler’s theorem, and draws the triangle with both circles.
How to use the circumcircle and incircle calculator
- Choose Three sides or Vertex coordinates.
- Enter sides a, b, c, or the coordinates of A, B and C.
- Pick the unit.
- Read the circumradius on the tape, with the inradius, both circles’ areas and circumferences, the triangle’s area and type, the distance between the centers and the ratio R/r. With coordinates you also get the circumcenter and incenter positions.
Formulas
With sides a, b, c, semi-perimeter s = (a + b + c)/2 and area K (from Heron’s formula or the shoelace formula):
For coordinates, the incenter is the weighted average (aA + bB + cC)/(a + b + c), where each vertex is weighted by the length of the side opposite it. The circumcenter is the point equidistant from all three vertices.
Worked example: the 13-14-15 triangle
Semi-perimeter: s = (13 + 14 + 15)/2 = 21.
Area: K = √(21 × 8 × 7 × 6) = √7,056 = 84.
Circumradius: R = 13 × 14 × 15 ÷ (4 × 84) = 2,730 ÷ 336 = 8.125.
Inradius: r = 84 ÷ 21 = 4.
Euler check: d = √(8.125 × (8.125 − 8)) = √1.015625 ≈ 1.008.
Placing the triangle at A(0, 0), B(14, 0), C(5, 12) gives the circumcenter (7, 4.125) and the incenter (6, 4); their distance is √(1 + 0.015625) ≈ 1.008 ✓.
The 13-14-15 triangle is a favorite because its area and both radii are whole numbers or simple decimals, and its 12-unit altitude splits it into a 5-12-13 and a 9-12-15 right triangle.
Quick reference for special triangles
| Triangle | Circumradius R | Inradius r |
|---|---|---|
| Equilateral, side s | s/√3 ≈ 0.577s | s/(2√3) ≈ 0.289s |
| Right, legs a and b, hypotenuse c | c/2 | (a + b − c)/2 |
| 3-4-5 right triangle | 2.5 | 1 |
| 5-12-13 right triangle | 6.5 | 2 |
For right triangles the circumcenter is the hypotenuse’s midpoint, so the circumradius is simply half the hypotenuse — a quick way to check any right-triangle result.
Practical uses
- Fitting a triangle into a circle, such as cutting a triangular part from round stock: the stock’s radius must be at least R.
- Fitting the largest circle into a triangle, such as a round fixture on a triangular plate: its radius is r.
- Locating a point equidistant from three sites — three towns, three sensors — which is the circumcenter.
- Mesh quality in engineering simulations, where the ratio R/r measures how far a triangle is from equilateral; 2 is perfect.
Related tools
Solve the triangle’s angles first with the triangle calculator or the law of sines calculator, whose ratio a/sin A equals 2R. The triangle area calculator explains Heron’s formula, and the circle equation calculator writes the circumcircle’s equation from the three vertices.
Frequently asked questions
What is the formula for the circumradius?
R = abc/(4K), where a, b and c are the sides and K is the area. Equivalently R = a/(2 sin A) from the law of sines. For a 13-14-15 triangle with area 84, R = 2730/336 = 8.125.
What is the formula for the inradius?
r = K/s, the area divided by the semi-perimeter s = (a + b + c)/2. For the 13-14-15 triangle, r = 84/21 = 4.
Where is the circumcenter?
At the intersection of the perpendicular bisectors of the sides. It lies inside an acute triangle, at the midpoint of the hypotenuse of a right triangle, and outside an obtuse triangle.
Where is the incenter?
At the intersection of the angle bisectors. It is always inside the triangle, and its coordinates are the vertices averaged with weights equal to the opposite side lengths.
How far apart are the two centers?
Euler's theorem gives d² = R(R − 2r). Since d² cannot be negative, R ≥ 2r for every triangle, with equality only for an equilateral triangle, where the centers coincide.