Triangle Solver

Triangle Solving: The Six Cases

A triangle has three sides (a, b, c) and three angles (A, B, C). To solve a triangle, you need at least three pieces of information — and at least one must be a side. The six 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)

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