Perimeter as the distance all the way round

0:00Draw any closed shape — any shape, as long as it comes back to where it started. Now put your finger on the edge and go all the way round, once, back to your starting point. The distance your finger travelled is the perimeter. That is the whole definition. And notice what it is not. Not how much room is inside — your finger never went inside. That confusion is the commonest mistake in this chapter, so let us fix it now.

0:31Perimeter is a journey round the edge. It is a length, and you could walk it. Now, if your shape is a polygon — a closed shape whose edges are all straight — something convenient happens. The boundary is just straight pieces, laid end to end. So instead of walking it, cut it at the corners and straighten it into one long line. The length of that line is the sum of the sides. A bent string is exactly as long as a straight one.

1:04Which is why the perimeter of a polygon is simply: add up the side lengths. That is not a rule to memorise. It is the same journey, done with arithmetic. And here is why anybody cares: every example in this section is something you buy. A tablecloth, three metres by two, and you want lace all the way round. How much lace? Three, plus two, plus three, plus two. Ten metres — and lace is sold by the metre.

1:37Or fencing a park a hundred and fifty metres by a hundred and twenty. Perimeter five hundred and forty metres, at forty rupees a metre — twenty-one thousand six hundred rupees. Lace, running, fencing — every one a length round an edge, charged by the metre. Now run it backwards. A rectangle has a perimeter of fourteen centimetres and a breadth of two. What is its length? The perimeter is two lots of length plus breadth. So length plus breadth is seven, and the length is five.

2:15Here is one with a small surprise. A rectangle of perimeter twelve metres, with a length of three. Length plus breadth is six, so the breadth is three. Three by three — it is a square, and the question was still telling the truth. And a trap the book plants on purpose. A field two hundred and thirty metres by a hundred and sixty, fenced with three rounds of rope.

2:42Three rounds needs three times the perimeter. Perimeter seven hundred and eighty, so two thousand three hundred and forty metres of rope. Take a piece of string thirty-six centimetres long, and bend it into shapes. A square first. Four equal sides, thirty-six divided by four — nine centimetres each. Now unbend it into an equal-sided triangle. Three sides, thirty-six over three — twelve centimetres each. Now a six-sided figure with equal sides. Thirty-six over six — six each.

3:22Three completely different shapes, and every one has a perimeter of thirty-six, because it is the same string. So the perimeter did not change when the shape did. Which should make you ask what did. Now two runners, on two tracks, one inside the other. The outer track is a rectangle, seventy metres by forty. One round of it is two hundred and twenty metres. The inner is sixty by thirty. One round, a hundred and eighty.

3:55Akshi runs five rounds of the outer track. Toshi runs seven rounds of the inner one. Who covers more ground? The obvious answer is wrong. Akshi: five rounds of two twenty is one thousand one hundred. Toshi: seven of one eighty is one thousand two hundred and sixty. Toshi — on the shorter track. She runs a hundred and sixty metres further, because distance is laps times perimeter, and she has more laps.

4:27A shorter loop does not mean a shorter run — and only one of those numbers is on the page. And a question you can only ask once you have a perimeter. Where is each runner standing? Akshi has run two hundred and fifty metres on a track two hundred and twenty round. So: one complete round, and thirty metres into her second. Toshi has run two fifty on a track of a hundred and eighty. One round, and seventy metres into her second.

5:01After a thousand metres, Akshi has done four rounds with a hundred and twenty left over. Toshi has done five rounds with a hundred. A division with a remainder, doing a useful job. The whole part is the laps; the remainder is where you stand. Here is the sharpest question in the section. Two square tracks sharing a centre. The inner has sides of a hundred metres; the outer, a hundred and fifty.

5:33So the inner track is four hundred metres round, and the outer is six hundred. Now one finish line through the middle of one side, crossing both tracks. And a race of exactly three hundred and fifty metres for both runners. Where does each of them start? You work backwards. Mark the finish, go three hundred and fifty metres back round your own track, and that is your start. And those two starting points are not next to each other. They cannot be — the tracks are different lengths, so three fifty reaches back a different fraction of each.

6:13A fair race on unequal tracks needs unequal starts. That is why athletics tracks stagger their starting lines. One small exercise before the last idea, and it is worth actually doing. Tear some shapes out of a newspaper. Random shapes — no straight edges, no formula, nothing you could look up. Estimate the distance round each, then lay a measuring tape round the edge and find out. Two things come of it. Your estimates get much better, quite quickly.

6:49And those torn shapes have a perimeter — a definite one, that a tape can find — though no formula applies to them. Now the argument the section ends on, and it is a real disagreement between two children in the book. There is a grid of dots, and on it a right-angled triangle. Three steps along the top, three down the side, then a sloping line back across three squares.

7:19Akshi counts. Three, plus three, plus three. Nine steps. She says the perimeter is nine units. Toshi says no. It cannot be nine. The book does not say who is right. It asks you to check — lay one flat step against one sloping step. Do that, and they do not match. The sloping step is longer — about one and four-tenths of a flat one. So Akshi counted nine steps, but three of them were the long kind. The real distance round is closer to ten and a quarter units.

7:56She was not slightly out. She was out by about twelve per cent, because she counted steps instead of measuring them. A step is only a unit if all the steps are equal. On this grid, they are not. So what do you write down? The book's answer is very good, and slightly surprising. You keep both units. You do not pick one. Call a flat step s, for straight. Call a sloping step d, for diagonal.

8:27Then that triangle is six s plus three d. Six flat steps and three sloping ones. And you must not collapse that into nine. That is the whole point of writing it this way. Six s plus three d is not unfinished. It is the complete answer, honest about something a single number hides. And here is the proof, from the book's own exercise. It gives four figures on that grid to measure. Two are worth putting side by side.

9:01The first comes to eight s plus two d. The second comes to four s plus six d. Now count the steps. Eight and two is ten. Four and six is ten. Both figures are ten steps round. So collapse each to a single number and both read ten units — and you would call their perimeters equal. They are not. A sloping step is about one and four-tenths of a flat one, so the first is about ten point eight units and the second twelve point four.

9:35The second figure is roughly fifteen per cent longer than the first. Same step count, genuinely different lengths. That is what collapsing throws away — and the book does not spell it out, but its own four figures prove it. So: perimeter is the distance round the edge. Add the sides when they are straight. Count in the right units. Next time, the other question this chapter asks — not how far round a shape is, but how much of the page it covers.