Enter any three elements of a triangle — three sides, two sides and an angle, or a side and two angles — and leave the rest blank. The calculator finds all sides, angles, area, altitudes, medians, angle bisectors and the radii of the incircle and circumcircle, and draws a diagram to scale with the solution steps.
How to solve a triangle
A triangle is fixed by three elements, at least one of which is a side. The classic cases:
- SSS — three sides: angles from the law of cosines.
- SAS — two sides and the included angle: the third side from the law of cosines.
- ASA / AAS — a side and two angles: the third angle from the 180° sum, sides from the law of sines.
- SSA — two sides and an angle opposite one of them: by the law of sines, with 0, 1 or 2 solutions.
Key formulas
- Law of cosines: c² = a² + b² − 2ab·cos C.
- Law of sines: a / sin A = b / sin B = c / sin C = 2R.
- Area: S = ½·ab·sin C; Heron: S = √(p(p−a)(p−b)(p−c)), p = (a+b+c)/2.
- Altitude h_a = 2S / a; median m_a = ½√(2b²+2c²−a²); angle bisector l_a = √(bc((b+c)²−a²)) / (b+c).
- Radii: r = S / p, R = abc / (4S).
When there are two solutions
Given sides a, b and an acute angle A opposite a: if b·sin A < a < b there are two triangles (angle B acute or obtuse); if a ≥ b there is one; if a = b·sin A there is one right triangle; if a < b·sin A there is none.
FAQ
How do I find the angles of a triangle from its three sides?
Use the law of cosines: cos A = (b² + c² − a²) / (2bc). That gives two angles; the third is 180° minus their sum. For sides 3, 4, 5 you get 36.87°, 53.13° and 90°.
How do I find the area of a triangle from three sides?
Use Heron's formula: S = √(p(p − a)(p − b)(p − c)), where p is the semi-perimeter. With two sides and the included angle it is simpler: S = ½·a·b·sin C.
Why do two sides and an angle sometimes give two triangles?
When the angle is opposite one of the known sides, the sine of the second angle is unique but the angle itself can be acute or obtuse (B and 180° − B). If both choices keep the angle sum below 180°, two different triangles exist.
Can a triangle be solved from three angles?
No. Three angles fix only the shape; there are infinitely many similar triangles. At least one side is required.
How do I find the inradius and circumradius?
Inradius r = S / p, circumradius R = abc / (4S). The table shows both, and you can toggle the circles on the diagram.