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Triangle Solver

Solve an SSS triangle from three side lengths to find its angles, perimeter and area with steps.

Calculate

SSS mode: enter all three positive sides in the same unit. This version does not support SAS, ASA, AAS or ambiguous SSA input.

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Formula and variables

cos A = (b² + c² − a²)/(2bc); p = (a+b+c)/2; area = √(p(p−a)(p−b)(p−c))

Sides a, b and c face angles A, B and C respectively. Angles are in degrees. p is the semiperimeter; area is in squared input units.

Worked example

For a = 3, b = 4, c = 5: cos A = (16 + 25 − 9)/40 = 0.8, so A ≈ 36.8699°. B ≈ 53.1301° and C = 90°. Perimeter = 12; p = 6; area = √(6 × 3 × 2 × 1) = 6.

When to use this formula

Use SSS when all three lengths are known. The triangle inequality must hold: the sum of any two sides is greater than the third. SSS fixes the size and shape, up to reflection.

Common mistakes

Do not confuse an opposite side with an adjacent side in the cosine rule. Inputs 1, 2, 3 describe a straight segment, not a triangle. Near-degenerate triangles can exceed floating-point precision.

Does SSS solve a geometry proof?

It computes measurements. A congruence proof also needs justified correspondences between two triangles; see the proof guide.

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