Law of Cosines Calculator

Solve triangles using the Law of Cosines for SSS and SAS cases with complete step-by-step solutions.

Select Case & Enter Values

Side a is opposite angle A, side b is opposite angle B, side c is opposite angle C.

Result

Triangle Solution
--
Side a--
Side b--
Side c--
Angle A--
Angle B--
Angle C--
Area--
Perimeter--

Step-by-Step Solution

c² = a² + b² - 2ab · cos(C)

Understanding the Law of Cosines

The Law of Cosines is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It generalizes the Pythagorean theorem, which is the special case when the angle is 90 degrees. The law is essential for solving triangles when you know three sides (SSS) or two sides and the included angle (SAS).

Law of Cosines Formulas

Find Side c

When you know sides a, b and the included angle C.

c² = a² + b² - 2ab·cos(C)

Find Side a

When you know sides b, c and the included angle A.

a² = b² + c² - 2bc·cos(A)

Find Side b

When you know sides a, c and the included angle B.

b² = a² + c² - 2ac·cos(B)

Find Angle C

When you know all three sides (SSS case).

cos(C) = (a² + b² - c²) / (2ab)

When to Use the Law of Cosines

  • SSS (Side-Side-Side): All three sides are known. Use the law to find any angle.
  • SAS (Side-Angle-Side): Two sides and the included angle are known. Use the law to find the third side, then find remaining angles.

For other cases (AAS, ASA, SSA), use the Law of Sines instead.

Relationship to the Pythagorean Theorem

When angle C = 90 degrees, cos(C) = 0, so the formula simplifies to c² = a² + b², which is exactly the Pythagorean theorem. The Law of Cosines is therefore a generalization that works for all triangles, not just right triangles.

Practical Applications

  • Surveying: Determining distances between points when direct measurement is difficult.
  • Navigation: Computing distances and bearings for air and sea travel.
  • Engineering: Analyzing forces in structural components and trusses.
  • Physics: Resolving vector addition problems.
  • Astronomy: Calculating distances between celestial objects.

Heron's Formula Connection

When you know all three sides (SSS), you can also calculate the area using Heron's formula: A = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2 is the semi-perimeter. This calculator automatically computes the area for you.