Area

Area is the amount of flat space a shape covers, measured in square units such as cm² and m². Count the squares, or use the formula for the shape; a compound shape splits into simpler shapes.

Key facts

  • Area is measured in square units: count the squares a shape covers.
  • Rectangle: area = length × width; perimeter = 2 × (length + width).
  • Triangle: area = ½ × base × perpendicular height.
  • Parallelogram: area = base × perpendicular height.
  • Trapezium: area = ½ × (a + b) × h, where a and b are the parallel sides.
  • Compound shapes: split into rectangles (or subtract a missing piece). Use the same units throughout.

Worked examples

Example 1

Count the squares to find the area of the shape. Each square is 1 cm².

  1. Count each square once. 1, 2, 3, 4, 5, 6, 7
  2. The area is 7 cm².

Answer: 7 cm²

Example 2

Find the area of a rectangle 5 cm long and 3 cm wide.

  1. There are 3 rows of 5 squares. 5 × 3 = 15 cm²

Answer: 15 cm²

Example 3

A rectangle is covered by 4 rows of 6 squares. Each square is 1 cm². What is its area?

  1. There are 4 rows of 6 squares. 6 × 4 = 24 cm²

Answer: 24 cm²

Example 4

Find the area of this L-shape. All corners are right angles; lengths are in cm.

  1. Draw a line to cut the L into two rectangles.
  2. The bottom rectangle is 10 cm by 4 cm. 10 × 4 = 40 cm²
  3. The top rectangle is 5 cm wide and 7 − 4 = 3 cm tall. 5 × 3 = 15 cm²
  4. Add the two parts. 40 + 15 = 55 cm²

Answer: 55 cm²

Example 5

A triangle has a base of 10 cm and a perpendicular height of 7 cm. Find its area.

  1. Draw a rectangle round the triangle: 10 cm by 7 cm. 10 × 7 = 70 cm²
  2. The height splits it into two smaller rectangles, and the triangle is exactly half of each one.
  3. So the triangle is half of the big rectangle. ½ × 10 × 7 = 35 cm²

Answer: 35 cm²

Example 6

A lawn 12 m by 8 m has a path 1 m wide all round it. Find the area of the path.

  1. The outside rectangle is 1 m bigger on every side. 12 + 2 = 14 m and 8 + 2 = 10 m
  2. Find its area. 14 × 10 = 140 m²
  3. Take away the lawn. 12 × 8 = 96 m²
  4. What is left is the path. 140 − 96 = 44 m²

Answer: 44 m²

Example 7

A trapezium has parallel sides of 6 cm and 10 cm and a height of 5 cm. Find its area.

  1. Halfway up, the width is the average of the parallel sides. ½ × (6 + 10) = 8 cm
  2. Cutting off the corners and turning them round makes a rectangle 8 cm wide and 5 cm tall with the same area.
  3. Area of that rectangle: 8 × 5 = 40 cm²

Answer: 40 cm²

Example 8

A trapezium has an area of 36 cm² and parallel sides of 5 cm and 7 cm. Find its height.

  1. Area = ½(a + b)h ½ × (5 + 7) × h = 36
  2. Solve. 6h = 36, h = 6 cm

Answer: 6 cm

Example 9

A circle has a radius of 5 cm. Find its area to 1 decimal place.

  1. Square the radius, then multiply by π. π × 5² = 78.539…
  2. Round to 1 decimal place. 78.5 cm²

Answer: 78.5 cm²

Example 10

Find the area of a parallelogram with a base of 8 cm and a perpendicular height of 5 cm.

  1. Area of a parallelogram = base × height 8 × 5 = 40 cm²

Answer: 40 cm²

Example 11

A trapezium has parallel sides of 7 cm and 11 cm and a height of 4 cm. Find its area.

  1. Add the parallel sides, halve, then multiply by the height. (7 + 11) ÷ 2 × 4 = 36 cm²

Answer: 36 cm²

Example 12

A right-angled triangle with shorter sides 4 cm and 3 cm is cut from a rectangle 12 cm by 5 cm. Find the area left.

  1. Rectangle: 12 × 5 = 60 cm²
  2. Triangle: ½ × 4 × 3 = 6 cm²
  3. Subtract. 60 − 6 = 54 cm²

Answer: 54 cm²

Example 13

A rectangle has an area of 48 cm² and a perimeter of 28 cm. Find its length and width.

  1. Half the perimeter is length + width. l + w = 14
  2. Find two numbers that add to 14 and multiply to 48. 8 + 6 = 14, 8 × 6 = 48

Answer: 8 cm, 6 cm