Perimeter of Rectilinear Shapes with Missing Sides

Perimeter is the distance all the way round a shape. A twelve by five rectangle: twelve plus five plus twelve plus five, thirty-four — or twice twelve plus five, because opposite sides are equal. A square of side seven is four sevens. Perimeter is a length, in centimetres or metres, never square units. Walk round the shape with your pencil and tick each side as you add it; the mistake with perimeters is a side missed, not a sum done wrong.

On a shape with square corners, some sides are usually unlabelled, and the rule for finding them is that the pieces going one way add up to the full side going that way. This L-shape is ten wide and eight tall. The two pieces across the top, four and six, add to the ten along the bottom; the two pieces down the right, three and five, add to the eight down the left. So a missing top piece is ten minus four, and a missing right piece is eight minus three. Then walk round: thirty-six. Which is the perimeter of the ten by eight rectangle the L fits in — an L-shape does not lose any perimeter by having a corner cut out.

A notch that goes into a side is different from a corner cut off. This shape is ten wide and eight tall with a notch four wide and three deep cut into the top. The top pieces still add up to ten — two, plus the four along the bottom of the notch, plus four — so the horizontal distance is the same as the rectangle's. But the two walls of the notch, three each, go down and back up, and they are extra. Perimeter: the rectangle's thirty-six plus two threes, forty-two. Walk round to check: it is the only way to be sure every wall is counted.

Perimeter in reverse: the perimeter is given and a side is wanted. A rectangle with perimeter fifty metres and long sides of fifteen. Half the perimeter is one long side plus one short side, twenty-five, so the short side is twenty-five minus fifteen, ten metres. Or the two long sides use thirty of the fifty, leaving twenty to share between the two short sides. With a relationship between the sides — length three times the width — write the perimeter in terms of the width: eight widths make eighty, so the width is ten.

The hardest version: a rectangle is joined to another and the perimeter changes. Two identical rectangles, twenty long, joined along a short side make one long rectangle, and the perimeter of the pair is not double the original, because the joined edge vanishes from both. Two perimeters would be one hundred and twenty-eight; the pair has one hundred and four; the twenty-four that vanished is twice the width, so the width is twelve. The general rule: add both perimeters and subtract twice the join. Draw it, and the change in perimeter is always a small number of side lengths — which is what finds the unknown one. Quiz and exercises now.