Triangle Solver

Triangle Solving: The Five Solvable Cases

A triangle has three sides (a, b, c) and three angles (A, B, C). To solve one you need three pieces of information, and at least one of them must be a side. The cases are named by what you know:

• SSS: Three sides known
• SAS: Two sides and the included angle known
• ASA: Two angles and the included side known
• AAS: Two angles and a non-included side known
• SSA: Two sides and a non-included angle known (the ambiguous case)

There is a sixth combination, AAA, and it is deliberately missing. Three angles fix the SHAPE of a triangle but not its size, so AAA describes an infinite family of similar triangles and there is nothing to solve. That is exactly why one of your three knowns has to be a length.

The Law of Cosines

Used when you know three sides (SSS) or two sides and an included angle (SAS):

a² = b² + c² − 2bc·cos(A)

b² = a² + c² − 2ac·cos(B)

c² = a² + b² − 2ab·cos(C)

Rearranged to find angles:

cos(A) = (b² + c² − a²) / (2bc)

Note: The Law of Cosines reduces to the Pythagorean theorem when C = 90°.

The Law of Sines

Used when you know two angles and any side (ASA or AAS), or two sides and a non-included angle (SSA):

a / sin(A) = b / sin(B) = c / sin(C)

All angles must sum to 180°: A + B + C = 180°.

The SSA Ambiguous Case

When given two sides and a non-included angle (SSA), there may be 0, 1, or 2 valid triangles:

• If the side opposite the given angle is shorter than the altitude from that angle, there is no triangle
• If it equals the altitude exactly, there is one right triangle
• If it is longer than the altitude but shorter than the adjacent side, there are two triangles
• If it is at least as long as the adjacent side, there is one triangle

Area Formulas

When base and height are known: Area = (1/2) × b × h

Heron’s Formula (when three sides are known):

s = (a + b + c) / 2
Area = √(s × (s−a) × (s−b) × (s−c))

When two sides and included angle are known: Area = (1/2) × a × b × sin(C)

Triangle Classification

By angles:

• Acute: All angles < 90°
• Right: One angle = 90°
• Obtuse: One angle > 90°

By sides:

• Equilateral: All three sides equal
• Isosceles: Two sides equal
• Scalene: All sides different

Real-World Applications

• Surveying: Find distances across impassable terrain
• Navigation: Calculate course and position using bearing angles
• Architecture: Roof pitch calculations, structural triangles
• Astronomy: Trigonometric parallax to measure stellar distances