The rectangle and square formulas are shortcuts for the same addition

0:00A rectangle. Twelve centimetres along the top, eight down the side. We already know how to find the perimeter — walk all the way round, and add up what you walked. So do that, and watch — a shortcut is about to fall out on its own. The corners are labelled A, B, C and D, so we can name the sides: A B along the top, B C down the right, C D along the bottom, D A up the left.

0:31Perimeter is A B plus B C plus C D plus D A. Four sides, added. Now, what do we know about a rectangle? Its opposite sides are equal. So C D is the same as A B — both twelve. And D A is the same as B C — both eight. So the sum is not four different numbers. It is twelve, plus eight, plus twelve, plus eight.

0:57Two twelves and two eights. The same two numbers, each turning up twice. And whenever something turns up twice, you can stop adding and start doubling. Two twelves is two times twelve. Two eights is two times eight. So the perimeter is two times twelve, plus two times eight. And both of those have a two in them. So take the two outside a bracket. Two times, in brackets, twelve plus eight.

1:28Twelve plus eight is twenty. Two twenties is forty. Forty centimetres. And there is your formula: the perimeter of a rectangle is two times length plus breadth — with the bracket. Now, two things about that formula which are worth being careful about. First, the bracket is not decoration. Two times, in brackets, length plus breadth means add first, then double. Without the bracket you have doubled only the length, and you get thirty-two, which is wrong.

2:03Second, and this one catches almost everybody. The perimeter is not length times breadth. Twelve times eight is ninety-six, and that is not the distance round anything. It is a different quantity, and we meet it two sections from now. Perimeter adds. Keep hold of that. And notice where the formula came from. Not from perimeter — from the rectangle. It works because a rectangle has two pairs of equal sides.

2:36Now a square photo frame, one metre a side, with coloured tape all round. Same question, same method. Walk round and add. One, plus one, plus one, plus one. Four metres of tape. That is the answer, and we did it the slow way on purpose. Because now look at the addition. One, four times over. Four of the same thing added is that thing times four. So the perimeter of a square is four times its side.

3:09And you can get there from the rectangle rule with no new thinking. A square is a rectangle whose length and breadth happen to be equal. Call them both s. The rectangle rule says two times, in brackets, s plus s. But s plus s is two s. So it is two times two s, which is four s. The square rule is not a second rule. It is the rectangle rule, with the two measurements the same.

3:39One idea — add the sides — wearing three different costumes. Two quick ones, using the rules forwards. A tablecloth three metres by two, lace all round. Two times, in brackets, three plus two. Two fives. Ten metres. And a square park with sides of seventy-five metres, run three times. One round is four times seventy-five, which is three hundred metres. Three rounds is nine hundred. Notice the second one used the square rule inside a bigger sum. The formula is a step, not the whole answer.

4:21Now backwards, which is where a rule earns its keep. A rectangle has perimeter fourteen and breadth two. Find the length. Two times, in brackets, length plus two, equals fourteen. So length plus two is seven. So the length is five. A square has a perimeter of twenty. Four times the side is twenty, so the side is five. And the best one in the set. A wire is bent into a rectangle five by three, then straightened and bent into a square. What is the square's side?

4:58The wire never got longer or shorter. Its length is the rectangle's perimeter — sixteen centimetres. Now that same sixteen is the square's perimeter. Four times the side is sixteen, so the side is four. Two rules had to agree about one piece of wire, and they did — because both are the same addition. But a rule running backwards has a limit, and it is worth seeing clearly. Suppose I tell you only that a rectangle's perimeter is twenty centimetres. What are its sides?

5:35You cannot say. It tells you length plus breadth is ten, and nothing more. One by nine. Two by eight. Three by seven. Four by six. Five by five. Five different rectangles, all with perimeter twenty — and more if you allow halves. So a perimeter fixes the sum of the two sides, and nothing else at all. Which should make you curious what does tell them apart. It comes later in this chapter.

6:07Now the part where the formulas stop working. Take a paper chit, six centimetres by four. Its perimeter is twenty. Cut it in half the long way. Two pieces, each six by two. Now rejoin them differently. End to end gives a long thin rectangle, twelve by two. Perimeter twenty-eight. Twenty-eight, not twenty. No paper was added, every piece kept its edges, and the boundary grew by eight centimetres. Now an L: one piece flat, the other hanging from its end. Twenty-eight again.

6:45Now a T: one standing upright on the middle of the other. Twenty-eight. Now stand them side by side, but slide one down by three centimetres. That one comes to twenty-six. Three of those came to exactly the same number, which looks like a coincidence and is not. Here is what is going on, and it explains all four at once. Each piece has a perimeter of sixteen. Two pieces, thirty-two centimetres of edge in total.

7:18Join them and some edge stops being outside — and it disappears twice, once from each piece. So the perimeter is thirty-two, minus two times the length they touch along. The first three all touched along a full two-centimetre end. Two times two is four. Thirty-two minus four is twenty-eight. All three. The fourth touched along three centimetres. Thirty-two minus six is twenty-six. And the book's closing puzzle solves itself. It wants a perimeter of twenty-two, so you need five centimetres of contact — stand them side by side and slide one down by one.

8:03One more thing falls out. The most they can touch is six, giving twenty — and that is the original chit, put back together. So cutting and rejoining can only make the boundary longer. Twenty is the floor, and the only way to reach it is to undo the cut. So notice what happened to our formulas in that last section. The L has no formula. The T has no formula. The offset pair has no formula.

8:36None is a rectangle, so the rule does not apply, and there is nothing to look up. But every one has a perimeter, and you found it the same way — round the edge, adding. That is the relationship between definition and formula. The definition always works; the formula is a shortcut the shape earns by repeating itself. A rectangle repeats twice, so you double. A square repeats four times, so you multiply by four. An L repeats nothing, so you add.

9:11One to go and do. Rule a border on a page of your own book — one centimetre in from top and bottom, one and a half from each side. Measure the page, take two centimetres off one direction and three off the other, then use the rule. The book prints no page size and the key skips it — the measurement is yours. Next time, the other question about a shape: not how far it is round the edge, but how much of the page it covers.