Why area is measured in squares and not in circles

0:00Every area you have counted so far, you counted in squares. Square units. Square centimetres. Square metres. Squares all the way down. And here the book stops and asks something almost nobody asks. Why squares? A girl in the margin puts it more sharply. Why not circles? Which is a completely fair question. Nobody ever justified the square to you. It was simply there. So let us take the question seriously, because the answer turns out to be about what a unit has to do.

0:38The book sets up an experiment, and it is worth actually doing. On a graph sheet, draw a circle whose diameter — the width straight across through the centre — is three. Now find its area, using the only method you have. Count the squares underneath. The whole squares in the middle are easy. Around the edge you use the rules: ignore the slivers, take the ones more than half covered.

1:06Do it carefully and you land somewhere near seven square units. Near seven. Not exactly seven, and you have no way at all of checking how close you got. Nothing you have been taught yet gives an exact area for a round region. That comes much later. But hold on to that number, because we are about to lose even that much certainty. So far so good. Now try the other side of the question.

1:36Instead of counting the squares the circle sits on, use circles as the counting unit itself. Take a rectangle and fill it with circles, all the same size, packed as tightly as you can. And look at what happens between them. Gaps. Four circles meeting leave a little pinched space in the middle that no circle reaches. Squares do not do this. Lay squares side by side and the region is covered — completely, with nothing left over and nothing doubled.

2:10Circles cannot manage that, and it is not a small imperfection. Packed in a plain grid, more than a fifth of the region is gap. And here is the figure the whole section turns on. The book prints one rectangle twice, filled with the same circles both times, arranged differently. On the left, a plain grid — circles in straight rows, each one directly above the one below. On the right, the rows are nudged sideways, so each row nests into the hollows of the row above.

2:48Count the left one and the book says forty-two. Count the right one and it says forty-four. Same rectangle. Same circles. Two different answers. Now the instinct is to say one of those must be a mistake. It is not. Both are correct counts of what is actually drawn. Forty-two circles really do fit that way. Forty-four really do fit the other way. And that is exactly the trouble. If the answer depends on how you arranged the units, then the answer is not a fact about the rectangle. It is a fact about your arrangement.

3:30The rectangle did not change. Your measurement did. A measurement that moves when the thing measured does not is not a measurement. And I checked this beyond the book's one figure. Nesting the rows is not even reliably better — in some rectangles the plain grid fits more, and in others the nested one does. Which is worse, not better. It is not that one packing is right. It is that the number is not a property of the region at all.

4:04Now, every class proposes the same rescue at this point, and it is a good idea, so let us take it seriously. Why not count the gaps as well? Add up the leftover slivers and include them. Try it, and notice what you have quietly done. You are no longer measuring in circles. You are measuring in circles and a second, differently shaped thing. Your unit has stopped being one shape, and a unit that is two shapes is not a unit.

4:36And the pinched gaps between the circles are exactly as hard to measure as the region you started with. You have not made progress. You have made the same problem, smaller. So write down what a shape has to do to be a unit of area. One. Copies of it must cover the region completely, leaving no gaps. Two. They must not overlap, or you count some space twice. That is the whole test, and it is a test about fitting together, not about being curved.

5:10That matters, because students often say circles fail because they are round. Roundness is not the problem. A shape could be curved on some edges and still tile perfectly, if the bumps of one fit the hollows of the next. Circles fail because the circle's bump has no matching hollow anywhere. Nothing fits against a circle. Now the book asks you to try other shapes against that test, and this is where it gets interesting.

5:44Try triangles. Take any triangle at all, and a copy of it turned upside down. The two together make a parallelogram, and parallelograms lay side by side with no gaps at all. So triangles pass. Completely, with no gaps and no overlaps, and not just special triangles — any triangle. That is a genuinely surprising fact, and it is worth pausing on. The unit did not have to be a square. Triangles would have worked.

6:18And rectangles obviously pass too. Bricks in a wall, tiles on a floor — you have seen it a thousand times. So now we have three shapes that all pass the covering test. Squares, triangles, rectangles. The covering argument tells you why circles are out. It does not tell you why the square is in. And here the book does something I respect. It asks you to make the list of the square's merits, and prints no list of its own.

6:50I looked for an answer in the rest of the chapter and there is not one anywhere. So what follows is my reasoning, not the book's. Judge it, do not memorise it. Here is the first reason, and I think it is the strong one. To pin down a rectangle as your unit, you have to state two numbers — how long and how wide. To pin down a triangle you need even more. Two sides and the angle between them, at least.

7:22To pin down a square you need one number. Its side. That is the whole description. So a square is the only one of the three that a single length fixes completely. And it makes a grid that is the same in both directions. Turn your squared paper a quarter turn and nothing about it has changed. Turn a grid of two-by-one rectangles a quarter turn and it is a different grid. Now which was the length and which the width?

7:54And here is the second reason, which is quieter but sits underneath everything you have done. You already have a unit of length. The centimetre. Take one centimetre, turn it through a right angle, and sweep it. What you get is a square of side one centimetre. And that is one square centimetre. The unit of area was manufactured out of the unit of length, with nothing added. Which is why the units in this chapter read the way they do. Square metres. Square centimetres. Square units.

8:30It is also why a rectangle's area comes out as one length multiplied by another at all. Length times breadth only produces an area because the unit you are counting in is itself a length by a length. Choose triangles as your unit and that sentence stops making sense. So here is the whole answer, in the order the argument actually runs. Circles are out because they leave gaps, and a unit that leaves gaps does not give the region a definite number at all.

9:07Triangles and rectangles are in — they genuinely tile, and either would have worked. The square wins on convenience, not on necessity. One number describes it, its grid looks the same in both directions, and it is built straight out of the unit of length. Those last three are my reasons. The book asks you for the list and leaves the page blank on purpose. So make your own list, and see whether you find one I have missed. That is a real invitation, not a rhetorical one.

9:43And two real jobs to finish on: find the floor area outside your corridor in square metres, and the area of your school playground. Both of them measured in squares — for reasons you can now actually give.