Working back from a perimeter

Now the problems turn round. You are told the perimeter and must find a missing side. Think of the perimeter as a piece of string laid all round the shape, and work backwards from its length.

A square has 4 equal sides, so each side is the perimeter ÷ 4. A square of perimeter 20 cm has side 20 ÷ 4 = 5 cm.

A rectangle's perimeter is 2 × (length + breadth), so half the perimeter is length + breadth. A rectangle of perimeter 50 m and length 15 m: half of 50 is 25, so length + breadth = 25 m, and the breadth is 25 − 15 = 10 m. In the same way, a rectangle of perimeter 14 cm has length + breadth = 7 cm; if the breadth is 2 cm, the length is 7 − 2 = 5 cm.

For a triangle, the three sides add up to the perimeter. A triangle of perimeter 55 cm with sides 20 cm and 14 cm has third side 55 − 20 − 14 = 21 cm.

The same string can make different shapes. A string 36 cm long makes:

• a square of side 36 ÷ 4 = 9 cm,
• a triangle with all sides equal, of side 36 ÷ 3 = 12 cm,
• or a hexagon (six equal sides) of side 36 ÷ 6 = 6 cm.

A wire bent into a rectangle 5 cm by 3 cm is 2 × (5 + 3) = 16 cm long. Straighten it and bend it into a square: each side is 16 ÷ 4 = 4 cm. The wire's length, the perimeter, stays the same; only the shape changes.