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Calculate Triangle Degrees

Reviewed by Calculator Editorial Team

Triangles are fundamental shapes in geometry with three sides and three angles that always add up to 180 degrees. Calculating triangle degrees is essential for solving problems in construction, navigation, and engineering. This guide explains how to find angles in triangles using the Law of Sines and Cosines, with practical examples and a free online calculator.

How to Calculate Triangle Degrees

Calculating triangle degrees involves determining the measures of the three interior angles. There are several methods depending on the information you have available:

When you know all three sides

Use the Law of Cosines to find one angle, then use the Law of Sines to find the other two angles.

When you know two sides and the included angle

Use the Law of Cosines to find the third side, then use the Law of Sines to find the other two angles.

When you know two angles and one side

First find the third angle (since angles in a triangle add up to 180°), then use the Law of Sines to find the other sides.

Important Note

All angles in a triangle must add up to exactly 180 degrees. If your calculations don't sum to 180°, there's likely an error in your measurements or calculations.

Formulas for Triangle Angles

Law of Sines

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

Where a, b, and c are the lengths of the sides opposite angles A, B, and C respectively.

Law of Cosines

c² = a² + b² - 2ab cos(C)

Or any permutation where c is the side opposite angle C.

The Law of Sines relates the sides of a triangle to the sines of its opposite angles. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.

Worked Examples

Example 1: Using the Law of Sines

Given a triangle with sides a = 5, b = 7, and angle A = 30°, find angle B.

  1. Use the Law of Sines: a / sin(A) = b / sin(B)
  2. 5 / sin(30°) = 7 / sin(B)
  3. 5 / 0.5 = 7 / sin(B)
  4. 10 = 7 / sin(B)
  5. sin(B) = 7 / 10 = 0.7
  6. B ≈ 44.43°

Example 2: Using the Law of Cosines

Given a triangle with sides a = 6, b = 8, and angle C = 60°, find side c.

  1. Use the Law of Cosines: c² = a² + b² - 2ab cos(C)
  2. c² = 6² + 8² - 2 × 6 × 8 × cos(60°)
  3. c² = 36 + 64 - 96 × 0.5
  4. c² = 100 - 48 = 52
  5. c ≈ √52 ≈ 7.21

FAQ

How do I know if my triangle angle calculations are correct?

The sum of all three angles in a triangle should be exactly 180°. If your calculations don't add up to 180°, double-check your measurements and calculations for any errors.

Can I use the Law of Sines for any triangle?

Yes, the Law of Sines can be used for any triangle, whether it's acute, right, or obtuse. It's particularly useful when you know two angles and one side or two sides and one angle.

What if I only know two sides and want to find the angles?

You'll need to know at least one angle to use the Law of Sines. If you don't know any angles, you can use the Law of Cosines to find one angle first, then use the Law of Sines for the others.