Area of a border or path

A rectangular garden is thirty-two metres long and eighteen metres wide. A path two metres wide runs all round the outside. Find the area covered by the path. Try it first. Pause if you like. The path is outside the garden. Does the larger length gain one border width, or one at each end? The blue guide spans the garden length. The purple parts show the same two-metre width added at the left and at the right. Length of the larger rectangle equals thirty-two metres plus two metres plus two metres, which equals thirty-six metres. The breadth also gains two metres at each end: one at the top and one at the bottom. Breadth of the larger rectangle equals eighteen metres plus two metres plus two metres, which equals twenty-two metres. Area of the larger rectangle equals thirty-six metres times twenty-two metres, which equals seven hundred and ninety-two square metres. Area of the garden equals thirty-two metres times eighteen metres, which equals five hundred and seventy-six square metres. The question asks for the path, not the garden. Which area must we subtract from the larger rectangle? Area of the path equals seven hundred and ninety-two square metres minus five hundred and seventy-six square metres, which equals two hundred and sixteen square metres. The area covered by the path is two hundred and sixteen square metres. The shaded border includes every corner once. Adding both ends gave the outside dimensions; subtracting the inside area leaves the path.

Mei Ling pastes a photograph on a card thirty centimetres long and twenty-four centimetres wide. A three-centimetre border remains all round. Find the area of the border. Try it first. Pause if you like. This time we know the outside dimensions. To find the photograph inside, do we add the border widths or subtract them? The card length includes the same three-centimetre border at the left and at the right. Both must come off to leave the photograph. Length of the photograph equals thirty centimetres minus three centimetres minus three centimetres, which equals twenty-four centimetres. The card breadth also includes three centimetres at the top and three at the bottom. Subtract both. Breadth of the photograph equals twenty-four centimetres minus three centimetres minus three centimetres, which equals eighteen centimetres. Area of the photograph equals twenty-four centimetres times eighteen centimetres, which equals four hundred and thirty-two square centimetres. Area of the card equals thirty centimetres times twenty-four centimetres, which equals seven hundred and twenty square centimetres. Area of the border equals seven hundred and twenty square centimetres minus four hundred and thirty-two square centimetres, which equals two hundred and eighty-eight square centimetres. The area of the border is two hundred and eighty-eight square centimetres. Change both ends of each dimension. Then subtract the inner area from the outer area.