Perimeter

Perimeter is the distance all the way around the outside of a shape. Imagine starting at one corner and walking right around the edge until you arrive back where you began — the total distance you walked is the perimeter. For this triangle with sides of 6, 8 and 10 centimetres, that trip is 6 plus 8 plus 10, which is 24 centimetres. Because it is a distance, the unit is an ordinary length unit like centimetres, never a squared one. Perimeter answers very practical questions: how much fence, ribbon or edging you need to go right around something. The mistake to guard against is losing track, so tick each side as you add it.

You can always add the sides one at a time, but rectangles and squares give you a shortcut. In a rectangle the opposite sides are equal, so for one that is 9 centimetres by 5, the long way is 9 plus 5 plus 9 plus 5, which is 28 centimetres. The shortcut says the same thing faster: add the length and the width, then double it. Nine plus five is fourteen, and doubling gives 28 centimetres. A square has all four sides equal, so its perimeter is just four times the side — a square of side 6 has a perimeter of 24 centimetres. Watch the bracket in the rectangle formula: you add first, then double, because both measurements need doubling.

When a shape is regular, every side is the same length, so instead of adding you can multiply. A regular hexagon with sides of 6 centimetres has six of them, giving 36 centimetres. An equilateral triangle with sides of 9 gives 27, and a regular pentagon with sides of 8 gives 40. Count the sides, multiply by the side length, done. The other thing to sort out before adding anything is units. If a rectangle is 1.2 metres by 40 centimetres you cannot add 1.2 to 40, because they are different things. Convert first: 1.2 metres is 120 centimetres, so the perimeter is two lots of 120 plus 40, which is 320 centimetres, or 3.2 metres.

Composite shapes are made by joining or cutting rectangles, and they usually arrive with a couple of sides unlabelled. Do not guess them — work them out. This L-shape is 12 centimetres wide overall, and the top edge is only 7, so the missing horizontal piece is 12 take away 7, which is 5. It is 8 centimetres tall and the short vertical is 3, so the missing vertical piece is 8 take away 3, also 5. Now add all six sides: 7 plus 3 plus 5 plus 5 plus 12 plus 8 gives 40 centimetres. A neat check: cutting a corner from a 12 by 8 rectangle swaps two edges for two equal ones, so 2 times 12 plus 8 also gives 40.

Questions often run the other way: you are told the perimeter and asked for a side. A square with a perimeter of 36 centimetres uses P equals four s, so 36 equals 4s and the side is 36 divided by 4, which is 9 centimetres. A rectangle takes one more step. If the perimeter is 30 and the length is 9, write P equals 2 times the bracket, substitute to get 30 equals 2 times 9 plus w, halve both sides to get 15 equals 9 plus w, and take 9 off both sides to leave w equals 6 centimetres. Always check by going forwards: 2 times 15 is 30, which matches. Write the formula, substitute, then undo the operations one at a time.