The cosine rule and the area of a triangle

The cosine rule handles triangles where the sine rule cannot start: when you know three sides, or two sides and the angle between them.

The cosine rule

a² = b² + c² - 2bc cos A. Rearranged for an angle:

cos A = (b² + c² - a²) / (2bc).

A negative cosine means the angle is obtuse.

Area = (1/2) ab sin C, where C is the angle between sides a and b.

In triangle ABC, b = 5 cm, c = 7 cm and A = 60°. Find a.

Answer: a = 6.24 cm (3 s.f.)

Key idea: a² = b² + c² - 2bc cos A; cos A = (b² + c² - a²)/(2bc); area = (1/2) ab sin C.

In triangle ABC, b = 8 cm, c = 11 cm and A = 50°. Find a to 2 decimal places.
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8.48 cm. a² = 64 + 121 - 176 cos 50° = 71.869, so a = 8.48 cm.
A triangle has sides 5 cm, 7 cm and 9 cm. Find its largest angle to 1 decimal place.
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95.7 °. The largest angle is opposite 9 cm. cos = (25 + 49 - 81)/70 = -0.1, so the angle is 95.7°.
A triangle has sides 3, 5 and 7. Find the exact cosine of its largest angle. Give your answer as a fraction in its simplest form.
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-1/2. cos = (9 + 25 - 49)/30 = -15/30 = -1/2.
Find the area of a triangle with sides 6 cm and 9 cm and an included angle of 40°. Give your answer to 1 decimal place.
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17.4 cm². Area = 0.5 x 6 x 9 x sin 40° = 17.4 cm².
A triangle has area 12 cm² and two sides 4 cm and 8 cm. Find sin C, where C is the angle between them.
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0.75. 12 = 0.5 x 4 x 8 x sin C = 16 sin C, so sin C = 0.75.
You know all three sides of a triangle and want an angle. Which rule do you use?
- The sine rule
- Area = (1/2) ab sin C
- Pythagoras' theorem
- The cosine rule
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The cosine rule. With three sides and no angle, only the cosine rule can start.

Last updated 7 November 2026. Spotted a mistake? Tell us.