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SQA National 5 Mathematics

Area of a triangle using sine

Using Area = 1/2 ab sin C for non-right-angled triangles.

Formula referenceMixed revision

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Check the triangle is not right-angled.
  • Identify two sides and the included angle.
  • Use consistent units.
National 5 Mathematics lesson

Explanation

The sine area formula finds the area of a non-right-angled triangle when two sides and the included angle are known.

The included angle is the angle between the two given sides.

This formula is often used when base and perpendicular height are not given directly.

Visual support

Included angle C

abAreaθ

Key formulae and rules

  • Area = 12 ab sin C

Watch out

Using an angle that is not between the two sides.

Check

Check your answer against the diagram. The longest side should still be opposite the largest angle.

Exam tip

If the angle is not between the two sides, do not use the sine area formula yet.

Calculator tip

Make sure the calculator is in degree mode before using sin, cos, tan or inverse trig.

Worked examples

Worked example 1

Find an area

Find the area of a triangle with sides 8 cm and 11 cm and included angle 40 degrees.

  1. Area = 12 ab sin C
  2. Area = 12 × 8 × 11 x sin 40
  3. Area = 28.28..

Answer: Area = 28.3 cm² to 1 decimal place

Worked example 2

Find a missing side

A triangle has area 30 cm², one side 10 cm and included angle 50 degrees. Find the other side.

  1. 30 = 12 × 10 x b x sin 50
  2. 30 = 5b sin 50
  3. b = 30 / (5 sin 50)

So: b = 7.8 cm to 1 decimal place

Worked example 3

Use exact substitution

Find the area when a = 12 m, b = 7 m and C = 90 degrees.

  1. Area = 12 × 12 × 7 x sin 90
  2. sin 90 = 1
  3. Area = 42

Answer: Area = 42 m²

Watch out

  • Using an angle that is not between the two sides.
  • Forgetting the 12
  • Writing square units incorrectly.