Area of any triangle

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Worksheet with answersDownload PDF

What this lesson covers

  • Area = ½ab sin C.
  • The angle must be between the two sides.
  • Finding an angle from an area can give two answers.
Key idea

The formula

  • Area = ½ab sin C, where C sits between sides a and b.
  • It is ½ × base × height, with b sin C as the height.
Worked example

Sides 8 and 11 with 35° between

  • Put the values straight in.
  1. A = ½ × 8 × 11 × sin 35°
  2. A = 44 × 0.5736
  3. A = 25.24 square units
Worked example

Working backwards

  • A triangle has area 30 with sides 8 and 10.
  1. 30 = ½ × 8 × 10 × sin C
  2. sin C = 0.75
  3. C = 48.6° or C = 131.4°

Common mistakes

  • Using an angle that is not between the two sides.
    Find the included angle first.
  • Giving one answer to sin C = 0.75.
    An obtuse angle has the same sine, so check both.

Your turn

  • 1. Find the area with sides 6 and 9 and 120° between them.
Show answer

1. ½ × 54 × sin 120° = 23.38 square units

Summary

What to take away

  • Two sides and the angle between gives the area.
  • Backwards, there may be two angles.

More in AS 91259

The sine and cosine rulesNotesArea of any triangleNotesBearings and 3D problemsNotesRadians, arcs and sectorsNotes

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