Triangle Calculator (SSS, SAS, ASA, AAS, SSA)

How it works

Law of cosines: c² = a² + b² − 2ab·cos C · Law of sines: a/sin A = b/sin B = c/sin C = 2R · A + B + C = 180° · Area = ½ab·sin C · r = Area/s, s = (a + b + c)/2

Three independent parts fix a triangle, as long as one of them is a side. SSS and SAS are solved with the law of cosines, ASA and AAS with the angle sum and the law of sines. SSA is the ambiguous case: sin B = b·sin A / a can give no triangle, one triangle or two (an acute and an obtuse B), and the calculator shows both when they exist. Side a is always opposite angle A, b opposite B, c opposite C.

Worked example

SSS with a = 5, b = 7, c = 8: cos B = (5² + 8² − 7²)/(2·5·8) = 0.5, so B = 60°; cos A = (7² + 8² − 5²)/(2·7·8) = 0.7857, so A = 38.2132° and C = 81.7868°. Missing values: A = 38.2132°, B = 60°, C = 81.7868°. Area = √(10·5·3·2) = 17.3205, perimeter 20, a scalene acute triangle.

Sources: OpenStax Algebra and Trigonometry 2e — 10.1 Non-right Triangles: Law of Sines · OpenStax — 10.2 Non-right Triangles: Law of Cosines

Frequently asked questions