Area of a triangle: ½ab sin C

The Maths Habit

Two sides of a triangle are 7 cm and 10 cm. Can you make its area bigger without changing those two sides? How?

Yes: change the angle between them. The area is biggest when it is 90°

Two sides are 14 cm and 12 cm, with 30° between them. Area = ½ × base × height. What is the height?

We know the base, 14 cm. We do not know the height h yet.

The pink right angled triangle has hypotenuse 12 cm and angle 30°.

h = 12 × sin 30° = 6 cm

Area = ½ × base × height

Area = ½ × 14 × 12 × sin 30° = 42 cm²

So Area = ½ × a × b × sin C

Area = 12ab sin C

Which angle do we need?

Sides of 7 cm and 10 cm The angle between them

Sides of 7 cm and 10 cm with an angle of 30° between them. Find the area.

12×7×10× sin30°
Area17.5 cm²

Sides of 8 cm and 5 cm with an angle of 60° between them. Find the area.

12×8×5× sin60°
Area17.3 cm²

Sides of 7 cm and 10 cm with an angle of 30° between them. Find the area.

12×7×10× sin30°
Area17.5 cm²

Sides of 6 cm and 9 cm with an angle of 110° between them. Find the area.

12×6×9× sin110°
Area25.4 cm²

Area of a triangle: ½ab sin C worksheet, page 1

Answer sheet

  1. A triangle has sides of 8 cm and 5 cm and an area of 10 cm². Find the acute angle between them.

    30°

  2. An equilateral triangle has sides of 6 cm. Find its area to 1 decimal place.

    15.6 cm²

  1. Find the area to 1 decimal place.

    17.4 cm2
    9 cm6 cm40°

  2. Find the area to 1 decimal place.

    34.4 cm2
    12 cm7 cm125°

  3. A triangle has an area of 30 cm². Two sides are 8 cm and 10 cm. Find the acute angle between them to 1 dp.

    48.6°
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