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½ ab sin C

06Area of a Triangle

Key idea

When you know two sides and the angle between them, the area follows in one line — no perpendicular height needed. The angle must be the one enclosed by the two sides you multiply.

The included angle

In ½·a·b·sin C, the angle C sits between sides a and b. If the angle you are given is not between the two sides, find another angle first.

Working backwards

If the area and two sides are known, rearrange to sin C = 2·Area / (ab), then take the inverse sine — remembering the obtuse possibility.

Right angles as a check

When C = 90°, sin C = 1 and the formula reduces to ½·base·height, a useful sanity check.

Area = ½ · a · b · sin C
C is the angle between sides a and b
Common error — Using an angle that isn't between the two chosen sides. The sine formula only works with the included angle.
Worked examples
Example 1 · Finding the area

A triangle has sides 12 cm and 9 cm with an included angle of 47°. Find its area.

  1. Area = ½ × 12 × 9 × sin 47°
  2. Area = 54 × 0.7314
  3. Area = 39.5 cm² (3 s.f.)
Example 2 · Finding an angle

Triangle XYZ has XY = 8 cm, XZ = 14 cm and area 40 cm². Find angle X.

  1. 40 = ½ × 8 × 14 × sin X
  2. 40 = 56 sin X
  3. sin X = 0.7143
  4. X = sin⁻¹(0.7143) = 45.6° (3 s.f.)

Exam-style question bank

5 of 5 shown
Foundations[3 marks]

A triangle has sides 10 cm and 7 cm with an included angle of 60°. Find its area.

Foundations[2 marks]

A triangle has two sides of 6 cm with a right angle between them. Find its area.

Standard[4 marks]

A triangle has sides 9 cm and 12 cm and an area of 50 cm². Find the two possible sizes of the included angle.

Standard[4 marks]

In triangle ABC, b = 10 cm, c = 15 cm and angle A = 40°. Find the area.

Exam[6 marks]

A triangular field ABC has AB = 50 m, BC = 65 m and angle ABC = 78°. Turf costs $3 per square metre. Find the cost of turfing the field.