Triangle Solver - Solve Any Triangle (SSS, SAS, ASA, AAS, SSA)

How to solve a triangle

A triangle has six elements: three sides (a, b, c) and three angles (A, B, C). You need at least three independent values to determine all six. Enter any three values above and the solver will compute the rest using the law of cosines and law of sines.

Supported cases

SSS: three sides. SAS: two sides and the included angle. ASA: two angles and the included side. AAS: two angles and a non-included side. SSA: two sides and a non-included angle (ambiguous case: may produce 0, 1, or 2 solutions).

Formulas used

Law of cosines: a² = b² + c² − 2bc·cos(A). Used to find a side when two sides and an included angle are known, or to find angles when all three sides are known.

Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Used when at least one angle-side pair is known.

Standard triangle diagram

          C
         /\
        /  \
       /    \
    b /      \ a
     /   A    \
    / α     γ  \
A ────────────── B
          c

Angles: A (α) at vertex A, B (β) at vertex B, C (γ) at vertex C
Sides: a opposite A, b opposite B, c opposite C
A + B + C = 180°

The ambiguous case (SSA)

When you know two sides and an angle that is not between them (SSA), there may be zero, one, or two valid triangles. Let h = b·sin(A) be the altitude from side a:

• a < h: the side is too short to reach the base - no triangle exists.
• a = h: the side is exactly the right length - one right triangle.
• h < a < b: two triangles are possible (the ambiguous case). The solver returns both solutions when they exist.
• a ≥ b: only one triangle is possible.

Area formulas

• Base × height: Area = ½ × base × height. The most familiar formula; requires a base and the perpendicular height.
• SAS (two sides and included angle): Area = ½ × a × b × sin(C). Useful when you have two sides and the angle between them.
• Heron's formula (SSS): Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. Works from three side lengths alone.

Real-world applications

• Surveying: triangulation uses triangle geometry to calculate distances and positions without direct measurement.
• Structural engineering: triangles are the only rigid polygon. Roof trusses, bridges, and towers rely on triangle geometry for load distribution.
• Navigation: bearing and distance calculations on sea and air routes use triangle-solving to determine position and heading corrections.
• Computer graphics: 3D meshes are composed entirely of triangles. Normal vector calculation, ray-triangle intersection, and texture mapping all use triangle math.