Triangle Solver (SSS, SAS, ASA, AAS, SSA)
Give any three parts of a triangle including a side and get all sides and angles, area, heights and circle radii — both triangles in the ambiguous SSA case.
A fully solved triangle with its type, drawn to scale, and the law (sines or cosines) that produced each value — or both solutions when the data allow two.
Example: Sides 7, 8 and 9 give angles 48.19°, 58.41° and 73.40°, an area of 26.83, inradius 2.236 and circumradius 4.696. Sides 7, 8 with A = 40° is ambiguous: B = 47.27° or 132.73°.
Three parts,
one triangle — usually.
Which cases exist, which law solves each, and why one of them can give two answers.
The five cases
Sides a, b, c lie opposite angles A, B, C. Three sides (SSS) give every angle from the law of cosines, cos A = (b² + c² − a²)/2bc. Two sides and the angle between them (SAS) give the third side from the same law. One side and two angles (ASA or AAS) give the third angle by subtraction from 180° and the other sides from the law of sines, a/sin A = b/sin B = c/sin C. Three angles alone are refused: they fix the shape, not the size.
The ambiguous case
Two sides and an angle not between them (SSA) is solved with the law of sines, and sin B = b·sin A / a has two angles with the same sine — B and 180° − B. If both leave a positive third angle, two different triangles have the given measurements and the page shows both; if sin B would exceed 1 there is no triangle at all. This is the one case where the input does not determine the answer, and it is flagged rather than silently picking one.
The other figures, and limits
Area comes from Heron's formula, √(s(s − a)(s − b)(s − c)) with s the semiperimeter; heights are 2·area / side; the inradius is area/s and the circumradius abc / 4·area. The type (acute, right, obtuse; scalene, isosceles, equilateral) is read from the largest angle and equal sides. Angles are in degrees; lengths are unit-free. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- Law of sines and law of cosines — standard trigonometric identities (any trigonometry text; e.g. Euclid II.12–13 for the law of cosines)
- Heron of Alexandria, Metrica, Book I — area from three sides
Last reviewed 19 September 2026. How results are checked: How we verify.