Why a triangle takes exactly half the rectangle around it

0:00Draw a rectangle on a piece of paper. Any rectangle. Now draw one diagonal — a straight line from one corner to the opposite corner. And cut along it. You now have two triangles. The book asks you to do something very simple with them, and it matters that you actually do it. Pick one triangle up, turn it over, and lay it on the other one. They cover each other. Exactly. Corner on corner, edge along edge, nothing sticking out anywhere.

0:34Now the book does something I want to point at, because it is a habit worth stealing. It does not stop there. It tells you to do it again with a different rectangle. A long thin one. And again with a short wide one. And again with a square. Every time, the two pieces coincide. That repetition is not padding. It is the difference between noticing something once and believing it.

1:02A relationship worth trusting is one that survives being tried again, on purpose, in cases you chose to be awkward. And then the book gives you a blank ruled line and asks you to write down what you found. It prints nothing on it. So write it. Two pieces, identical, and together they are the whole rectangle. If two equal things make a whole, each one is half of it. There is nothing more to the step than that.

1:35So a triangle cut off a rectangle by its diagonal has exactly half the rectangle's area. Not about half. Not roughly. Half. And notice you got there without measuring anything at all — no lengths, no counting, no formula. You got there by lifting a piece of paper and putting it down on another one. The book then shows you two shapes, and asks a question it refuses to answer.

2:05A rectangle with a diagonal across it. And beside it, a triangle with a line dropped from its top corner to its base. Is the rectangle's area more than the triangle's, less, or the same? Both answer lines are printed blank. So we are not going to settle it by eye. Look at the shapes though — the triangle's base is roughly twice the rectangle's length, and they stand about the same height.

2:34Which should make you suspicious that the answer is more interesting than it looks. We will settle it by counting. So onto grid paper, where counting settles things exactly. Rectangle A B C D. A at the bottom left, B bottom right, C top right, D top left. Five squares across, four squares down. Twenty square units. No estimate — every corner is on a grid point. Now mark a point E on the top edge, three squares along from D.

3:10And mark F on the bottom edge, directly below E. The line from E down to F splits the rectangle into two smaller rectangles. Remember them — they do all the work later. First, the easy triangle. Triangle B A D — corner to corner to corner. Its top corner sits directly above a corner of the rectangle, so this is exactly the case we already settled. It is the rectangle cut by its diagonal. So its area is half of twenty. Ten square units.

3:48And you can check that by counting squares on the grid, which is worth doing once. Ten. Which agrees, as it had better. Now the interesting one. Triangle A B E. Same base — the whole bottom edge, A to B. Same height — E is on the top edge, so it is just as high up. But the top corner has moved. E is not above a corner. It is three squares in.

4:16So the diagonal argument does not apply. This triangle is not half a rectangle cut by its diagonal — it is not a diagonal of anything. And here is where a lot of people would reach for a new rule. The book does not. It uses the old one twice. Drop the line from E straight down to F, and look at what it did to the triangle. It cut triangle A B E into two triangles. A E F on the left, and B E F on the right.

4:51Now look at A E F. Its corner at F is a square corner, and it sits inside the left rectangle, A F E D. In fact it is exactly that rectangle cut along its diagonal. So A E F is half of A F E D. And B E F sits inside the right rectangle, F B C E, and is exactly that one cut along its diagonal. So B E F is half of F B C E.

5:21Two triangles we could not do, turned into two we already had. So put it together, and watch the whole thing fall out. Triangle A B E is A E F plus B E F. That is half of the left rectangle, plus half of the right rectangle. And half of one thing plus half of another is half of the two put together. The two rectangles put together are the whole rectangle A B C D.

5:51So triangle A B E is half of A B C D. Half of twenty. Ten square units. On the numbers: the left rectangle is three by four, twelve, so A E F is six. The right is two by four, eight, so B E F is four. Six plus four is ten. Now put the two triangles side by side, and look at what you have just proved. Triangle B A D — a right-angled triangle, leaning hard to one side. Ten square units.

6:26Triangle A B E — a symmetric-looking one with its peak part way along. Ten square units. They look nothing alike. They are the same size. And now the really valuable bit. Slide E along the top edge. Anywhere you like. Put it above A. Put it above B. Put it a quarter of the way along, or seven-eighths. Every single one of those triangles is ten square units, because the argument never used where E was.

6:59The base did not change and the height did not change, so nothing changed. That is the idea to keep. The book then prints two blank ruled lines and asks you to write the conclusion. It writes nothing itself. So here it is. A triangle takes exactly half of the rectangle built on its base and reaching its height. And one warning that belongs with it, because the statement is easy to over-read.

7:30Not any rectangle you happen to draw round the triangle. That one. Draw a rectangle that is taller than the triangle, or wider than its base, and the triangle is no longer half of it. Count the squares and you will see it fail. Notice what the chapter never does, either. It never writes a formula. No base times height divided by two, anywhere. It did not need one. If you followed the argument you can rebuild the answer for any triangle you meet — and the argument is the thing worth having.

8:09And now you can measure things that have no rule at all. The book ends with five figures on a grid. Slanted four-sided ones, a five-sided one with a peak, another with a notch cut out of its top. None of them is a rectangle. None is a triangle. Not one has a formula. But every one falls apart into rectangles and triangles, and you can do both. Cut, work out each piece, and add: twenty-four square units. Thirty. Forty-eight. Sixteen. Twelve.

8:45That is the pattern this whole chapter has been building. Count when you can, cut when you cannot, and add. Rectangles gave you the counting. The diagonal gave you the triangle. Between them, they cover every straight-sided figure you will ever be handed. Next time we leave measurement altogether, and start on fractions.