Area as a count of unit squares

0:00Here is a square, cut into four equal parts. One cut down, one cut across, and nobody argues. Now here is the same square cut a different way. The boundaries step in and out. No piece is a square any more, and no two pieces are even the same shape. Are those four parts still equal? Most people hesitate, and the hesitation is the interesting part. It means we are not sure what equal is supposed to mean.

0:34Watch one piece. A bite is pushed out of it along one edge - and exactly the same bite is pushed in along the next. It lost some material and gained the same amount back. So whatever quantity we are measuring, that piece still has as much of it as it started with. Every piece does the same trade, so all four stay equal. And nothing stopped us choosing a different bite, so there are not two ways to do this. There are as many as you have patience for.

1:07But notice what we just leaned on: that the quantity survives being moved. Before we can trust that, we have to say what the quantity is. So here is a question that will force us to say it. Two shapes are to be painted, evenly, in the same colour. One is seven across and four down. The other is eight across and three down. Which one needs more paint? You cannot answer that yet, and not because the arithmetic is hard. There is nothing here to count. We have two lengths each, and paint is not a length.

1:46The move is to choose something to count with. Take a square whose side is one unit, and ask how many of them each shape would hold. Lay that square down again and again across the first shape. Seven of them fit along the bottom. That row of seven repeats four times going up. Twenty-eight squares. The second shape takes eight along the bottom, and that row repeats three times. Twenty-four.

2:16So the first one needs more paint, and now we know what more means. It holds four more of the squares we chose. Hold on to one detail here, because everything after this comes out of it: the squares did not arrive one at a time. They arrived in rows, and every row was the same length. That count has a name. The area of a shape is how many unit squares its material would fill.

2:46Fractions are allowed, and they have to be. Nothing says a shape will hold a whole number of squares, and a definition that only worked when it did would be no use at all. Notice what area is a count of. Not the edge, not the outline - the material. And notice what it is not. It is not a formula. It is a number of squares, and it exists before anyone knows how to work it out quickly.

3:16Now for working it out quickly. Because the squares came in equal rows, we never had to tally them. Four rows of seven is seven fours. Length times width. That is worth saying carefully. Length times width is not what area means. It is a claim about how the squares are arranged in a rectangle, and it is true only because they arrive in equal rows. It also tells you why the unit carries a little two. Twenty-eight square centimetres names a square of side one centimetre, twenty-eight times over.

3:55Which is why a square metre is not a hundred square centimetres. Lay centimetre squares along a metre and you get a hundred. Then repeat that row a hundred times. Ten thousand. Draw a diagonal across the first rectangle. Two triangles. Lift one and lay it on the other and they match exactly, so they hold the same amount of material. Two equal pieces making twenty-eight between them, so each is fourteen.

4:29Look at what the diagonal did to the squares, though. Only nine of the twenty-eight lie whole below the line. The line cuts straight through ten of them. The pieces of those ten come to exactly five, which is why the answer is fourteen and not nine. The squares did not cooperate, and the count did not need them to. That is the fractions clause earning its place, the first time we have used it.

5:00If area is a side times a side, then an area divided by one side gives the other. That runs figures backwards. Here are four rectangles arranged around a centre. All four areas are marked. Four short lengths are marked. Every other length is missing, including the one we want. There is exactly one place to start. Twenty-one over seven is three, so that rectangle is three tall. Three and four is seven. Twenty-eight over seven is four. Four and three is seven. Thirty-five over seven is five. Five and two is seven. Fourteen over seven is two.

5:45Two. And every step was one division or one addition. Nothing was measured and nothing was guessed. At each stage exactly one length became available, which is what makes it an argument. Not every figure is that obliging. Here is one that is not. A tall rectangle. Its upper band is marked twenty-nine, and that band is four high. Beside it sits a narrow rectangle marked eleven, level with the band. And a fifty is written against the left edge.

6:22Two of the three answers come out whatever the fifty belongs to. Twenty-nine over four is seven and a quarter. Eleven over four is two and three quarters. The third one depends. If the fifty is the whole rectangle, the blank part below holds twenty-one, and its height is twenty-one over seven and a quarter. If the fifty is that blank part alone, its height is fifty over seven and a quarter. Those are different numbers, and the figure does not say which.

6:58So we say both. Which label constrains which length is the lesson of a figure like this, and pretending it came out neatly throws that away. Back to the promise we made at the start. A rectangular park sits inside a rectangular field, and the strip left over is a path. The field holds eighty squares, the park holds twenty-four, so the path holds fifty-six. Now slide the park. Push it to a corner, push it to an edge, put it anywhere with the path still running all the way round. Count the path again.

7:38Fifty-six, every time. Of course it is - the field did not change and the park did not change, so what is left over cannot have changed either. You can also cut the path into four strips and add them up. Same fifty-six. The pieces of a shape always total the shape, and that is the property doing the work in every argument here. There is one way this goes wrong, and it goes wrong twice.

8:09A plot fourteen by twelve, with a path straight across it and another straight down. The one across is two wide, so it holds twenty-eight. The one down is three wide, so it holds thirty-six. Add them: sixty-four. But the true count is fifty-eight. The six squares where the two paths cross belong to both, and they were counted twice. The same thing happens to a bent tube one square wide, with two arms of five. Five and five is ten, and the tube holds nine. The corner square is in both arms.

8:48Which is worth more than the correction. If a bent tube of nine squares is a straight tube nine long, then any tube of constant width can be straightened before it is measured, corner by corner. One last figure, and it is a trap. A square is cut into four pieces by two straight cuts. The pieces are shuffled and laid out again - and the square they make is bigger than the one we started with.

9:18Nothing was created. Look at the middle: there is a hole in it now. The four pieces hold exactly what they always held, and the hole is precisely the difference between the two squares. That is the same argument as the very first one. A shape can be cut up, moved, bent, straightened and laid out wider, and the count of its unit squares never notices, because the material never changed.

9:47Area is a count. Length times width is a shortcut that a rectangle happens to allow, and every shape after the rectangle is somebody finding a way to move the material until it counts as easily.