What is the law of cosines?

The law of cosines generalises Pythagoras to any triangle: c² = a² + b² − 2ab·cos C. When C = 90°, cos C = 0 and the formula reduces to c² = a² + b², recovering the Pythagorean theorem. It works for all triangles, not just right-angled ones.

When do I use the law of cosines instead of the law of sines?

Use the cosine rule when you know all three sides (SSS) or two sides and the included angle (SAS). Use the sine rule for ASA, AAS, or to resolve ambiguous SSA cases after finding an initial angle with the cosine rule.

What is the SSA ambiguous case?

When you know two sides and a non-included angle (SSA), there can be zero, one, or two valid triangles depending on the relative lengths. The solver checks sin B = b·sin A / a: if this exceeds 1 there is no solution; if it equals 1 there is exactly one (a right triangle); otherwise it tries both the acute and the obtuse value of B and discards any that give a negative third angle.

How is the triangle area calculated?

Once two sides and the included angle are known, the area formula ½·a·b·sin C applies. For SSS input the solver first finds an angle via the cosine rule, then uses that formula. The result is in square units matching whatever units you used for the sides.

Can I use any units — metres, feet, or abstract values?

Yes. The calculator is unit-agnostic. Sides in metres give area in m², sides in feet give area in ft², and dimensionless side lengths (for pure geometry problems) work just as well. Angles are always in degrees.

Why does the solver sometimes return only one SSA solution instead of two?

Two solutions only exist when both candidate angles B produce a positive third angle C = 180° − A − B. If the obtuse candidate (B = 180° − arcsin(sin B)) would make C zero or negative, that case is geometrically impossible and the solver suppresses it, leaving exactly one valid triangle.

Law of Cosines Solver

The law of cosines solver finds every unknown side, angle, area, and perimeter of any triangle — right-angled or oblique — given just three pieces of information. It covers all five standard congruence cases (SSS, SAS, ASA, AAS, SSA) and, uniquely, handles the SSA ambiguous case by detecting whether zero, one, or two valid triangles exist and showing the full algebraic reasoning for each.

The formula

The law of cosines states:

c² = a² + b² − 2ab · cos C

Here side a is opposite angle A, side b opposite B, and side c opposite C. Rearranging to find an angle:

cos C = (a² + b² − c²) / (2ab)

When angle C is 90° the cosine term vanishes and the formula collapses to the Pythagorean theorem, confirming that the cosine rule is the general form. The companion law of sines (a / sin A = b / sin B = c / sin C) is used internally for the ASA and AAS cases and to resolve the ambiguous SSA scenario.

How it works

SSS (three sides known): The solver applies the rearranged cosine rule to find angle A from sides a, b, c, then B from the same formula, and finally sets C = 180° − A − B. Area follows from ½ · a · b · sin C.

SAS (two sides + included angle): With sides a, b and included angle C, the cosine rule gives c directly (c² = a² + b² − 2ab · cos C). The remaining angles A and B then come from the cosine rule applied to the completed side set.

ASA / AAS (two angles + one side): The third angle is C = 180° − A − B, then the sine rule propagates to find the remaining sides.

SSA (ambiguous): The solver computes sin B = b · sin A / a. If sin B > 1 there is no triangle. If sin B = 1 there is exactly one right triangle. Otherwise both B = arcsin(sin B) and B = 180° − arcsin(sin B) are tested; any candidate that yields a negative C is discarded. One or two valid triangles are displayed side by side.

Worked example

Suppose you know all three sides of a surveyor’s triangle: a = 7 m, b = 8 m, c = 9 m (SSS).

cos A = (8² + 9² − 7²) / (2 × 8 × 9) = (64 + 81 − 49) / 144 = 96 / 144 = 0.6667 → A ≈ 48.19°
cos B = (7² + 9² − 8²) / (2 × 7 × 9) = (49 + 81 − 64) / 126 = 66 / 126 = 0.5238 → B ≈ 58.41°
C = 180° − 48.19° − 58.41° = 73.40°
Area = ½ × 7 × 8 × sin(73.40°) = ½ × 7 × 8 × 0.9583 ≈ 26.83 m²
Perimeter = 7 + 8 + 9 = 24 m

All five of these values appear instantly when you type 7, 8, 9 into the SSS inputs above.

Formula reference

Case	Given	Primary formula
SSS	a, b, c	cos A = (b² + c² − a²) / (2bc)
SAS	a, b, C	c² = a² + b² − 2ab·cos C
ASA	A, B, c	C = 180−A−B; sides via sine rule
AAS	A, C, a	B = 180−A−C; sides via sine rule
SSA	a, b, A	sin B = b·sin A / a (0, 1, or 2 triangles)

Every calculation runs locally in your browser — no data is transmitted to any server.