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SAS · SSS

05The Cosine Rule

Key idea

The cosine rule covers the two cases the sine rule cannot reach: two sides and the included angle (to find the third side), and three sides (to find any angle). It is Pythagoras with a correction term for the angle.

Finding a side (SAS)

Given two sides and the angle between them, use a² = b² + c² − 2bc·cos A, where A is the included angle and a is the side opposite it.

Finding an angle (SSS)

Given all three sides, rearrange to cos A = (b² + c² − a²) / (2bc). The angle you find is opposite side a.

Obtuse angles are automatic

Unlike the sine rule, cos⁻¹ returns the correct angle directly — a negative cosine gives an obtuse angle with no ambiguity.

a² = b² + c² − 2bc·cos A
cos A = (b² + c² − a²) / (2bc)
Common error — Using the cosine rule with a non-included angle. The angle must sit between the two sides you square.
Worked examples
Example 1 · Finding a side (SAS)

In triangle ABC, b = 7 cm, c = 10 cm and angle A = 55°. Find side a.

  1. a² = 7² + 10² − 2(7)(10)cos 55°
  2. a² = 49 + 100 − 140 × 0.5736
  3. a² = 149 − 80.3 = 68.7
  4. a = 8.29 cm (3 s.f.)
Example 2 · Finding an angle (SSS)

A triangle has sides 6 cm, 9 cm and 11 cm. Find the largest angle.

  1. The largest angle is opposite the longest side (11 cm).
  2. cos A = (6² + 9² − 11²) / (2 × 6 × 9)
  3. cos A = (36 + 81 − 121) / 108 = −0.0370
  4. A = cos⁻¹(−0.0370) = 92.1° (3 s.f.)

Exam-style question bank

5 of 5 shown
Foundations[3 marks]

In triangle ABC, b = 5 cm, c = 8 cm and angle A = 60°. Find side a.

Standard[4 marks]

Two sides of a triangle are 8 cm and 11 cm with an included angle of 35°. Find the third side.

Standard[4 marks]

A triangle has sides 7 cm, 8 cm and 9 cm. Find the smallest angle.

Exam[5 marks]

A parallelogram has sides 10 cm and 6 cm with an angle of 70° between them. Calculate the length of the shorter diagonal.

Challenge[6 marks]

A triangle has sides a = 13 cm, b = 14 cm and c = 15 cm. Find all three angles.