0580

Sine Rule, Cosine Rule and Area of Triangles

Pythagoras and Trigonometry

Two sides and their included angle

  • The printed formula is Area = ½ab sin C
  • Angle C must sit between the two sides a and b, and any pair of sides with their included angle will do
  • Where that angle is not given, find it first with the sine or cosine rule
  • If C happens to be 90°, sin 90° = 1 and the formula collapses to the familiar half base times height
  • Convert all the lengths to the same unit before substituting, and give the area in the unit the question asks for
Triangle ABC marked not to scale with AC given as 1.1 m, AB as 32 cm and the angle at A between them as 74 degrees, so the two sides and their included angle are known but the units must be matched before the area formula is used
Source: Area of a triangle by Save My Exams
Exam tip

Match each side to the angle opposite it before substituting. Use the sine rule where a side and its facing angle are both known, and the cosine rule where they are not. Inverse sine returns only the acute angle, so check whether 180° minus it is the one the diagram shows — the syllabus includes that ambiguous case deliberately. The cosine rule has no such ambiguity.

Worked example

Area from two sides and the included angle

A triangle has sides of 8.5 cm and 6.2 cm with an angle of 63° between them. Find its area, correct to 3 significant figures.

Solution:

  • The angle lies between the two given sides, so the area formula applies directly
  • Area = ½ × 8.5 × 6.2 × sin 63°
  • Area = 26.35 × sin 63°
  • Area = 23.4780…
  • The area is 23.5 cm² to 3 significant figures