Perimeter and area

The perimeter is the total distance around the outside of a shape.

1. Find any missing side lengths, using the lengths you know.
2. Add all the sides, going round the shape once.

Example: an L-shape is made by cutting a 4 cm by 3 cm rectangle from one corner of a 10 cm by 8 cm rectangle. Find its perimeter.

1. Going round, the sides are 10 cm, 8 − 3 = 5 cm, 4 cm, 3 cm, 10 − 4 = 6 cm and 8 cm
2. 10 + 5 + 4 + 3 + 6 + 8 = 36 cm

Cutting a rectangle from a corner doesn't change the perimeter: it's still 2 × (10 + 8) = 36 cm.

Area is the space inside a shape, in square units such as cm².

Example: find the area of a trapezium with parallel sides of 6 cm and 10 cm, 4 cm apart.

1. ½ × (6 + 10) × 4
2. = ½ × 16 × 4
3. = 32 cm²

To find the area of a shape made of simpler shapes:

1. Split it into rectangles and triangles (or treat it as a big shape with a piece cut out).
2. Find any missing lengths.
3. Find each area, then add (or subtract the cut-out piece).

Example: find the area of the L-shape from the perimeter section: a 10 cm by 8 cm rectangle with a 4 cm by 3 cm corner cut out.

1. Big rectangle: 10 × 8 = 80 cm²
2. Cut-out: 4 × 3 = 12 cm²
3. 80 − 12 = 68 cm²

Check by splitting instead: 6 × 8 = 48 and 4 × 5 = 20, and 48 + 20 = 68.

When a length is unknown, write the area in terms of a letter and form an equation.

Example: a triangle has an area of 30 cm² and a base of 12 cm. Find its height.

1. ½ × 12 × h = 30
2. 6h = 30
3. h = 5 cm

Harder example: a rectangle is (x + 3) cm long and (x − 1) cm wide. Its area is 32 cm². Find its length and width.

1. (x + 3)(x − 1) = 32
2. x² + 2x − 3 = 32, so x² + 2x − 35 = 0
3. (x + 7)(x − 5) = 0, so x = 5 (a length can't be negative)
4. 8 cm long and 4 cm wide

Check: 8 × 4 = 32.