Last updated: 2026-05-19

The law of cosines

The law of cosines relates the lengths of all three sides of a triangle to the cosine of one of its angles. For a triangle with sides a, b, c and the angle C opposite side c, the formula is:

c² = a² + b² − 2ab cos(C)

This single equation covers two classic triangle-solving scenarios:

• SAS (Side-Angle-Side): two sides and the angle between them are known. The law of cosines gives the third side.

• SSS (Side-Side-Side): all three sides are known. Rearranging the formula three times yields all three angles:

  A = arccos((b² + c² − a²) / (2bc))
  B = arccos((a² + c² − b²) / (2ac))
  C = arccos((a² + b² − c²) / (2ab))

How to use this calculator

SAS mode — select "SAS — Find Side c", enter sides a and b and the angle C between them (in degrees). The calculator returns side c.

SSS mode — select "SSS — Find Angles", enter all three side lengths a, b, c. The calculator returns all three angles A, B, C in degrees.

Worked example: SAS

A surveyor knows two boundary lengths of a triangular plot — 83 m and 112 m — and the angle between them is 54°. What is the third boundary length?

• a = 83, b = 112, C = 54°
• c² = 83² + 112² − 2 × 83 × 112 × cos(54°)
• c² = 6 889 + 12 544 − 18 592 × 0.5878
• c² = 19 433 − 10 931 ≈ 8 502
• c ≈ 92.2 m

Worked example: SSS

A triangular garden has sides 5 m, 7 m, and 6 m. What are the corner angles?

• A = arccos((49 + 36 − 25) / (2 × 7 × 6)) = arccos(60/84) ≈ 44.4°
• B = arccos((25 + 36 − 49) / (2 × 5 × 6)) = arccos(12/60) ≈ 78.5°
• C = arccos((25 + 49 − 36) / (2 × 5 × 7)) = arccos(38/70) ≈ 57.1°

Check: 44.4° + 78.5° + 57.1° = 180°, confirming the three angles are consistent.

Why the formula works: connection to the Pythagorean theorem

The Pythagorean Theorem Calculator (c² = a² + b²) is a special case of the law of cosines. When angle C is exactly 90°, cos(90°) = 0 and the 2ab cos(C) term vanishes, leaving c² = a² + b².

For any other angle:

• If C < 90° (acute), cos(C) > 0, so the subtracted term makes c shorter — the triangle "closes up".
• If C > 90° (obtuse), cos(C) < 0, so the term becomes an addition, and c is longer than the hypotenuse would be.

This geometric intuition comes directly from the dot product definition of cosine, which measures how much one vector projects onto another.

Law of cosines vs. law of sines

The law of sines is algebraically simpler when you already have an angle–opposite-side pair. The law of cosines has no ambiguous case and is preferred when the initial data is purely sides-and-included-angle.

Triangle inequality

For three lengths to form a valid triangle, every side must be less than the sum of the other two. If a side is ≥ the sum of the remaining two (for example, sides 1, 2, and 10), the arccos argument falls outside [−1, 1] and is undefined. The calculator shows a validation error in this case. Arrange sides in increasing order and verify that the two smaller ones sum to more than the largest.