Why the area of a triangle is half base times height

0:00Here is a rectangle, five across and four up. Draw a triangle inside it: the rectangle's bottom side is the triangle's bottom side, and its top corner sits somewhere along the top edge. Now slide that top corner along. The triangle changes shape completely. It leans left, it stands upright, it leans right. The area does not move. Ten, ten, ten, ten, ten. And ten is exactly half of twenty, which is what the rectangle holds.

0:36So every one of those triangles is half the rectangle. Why should sliding that corner change nothing? Drop a perpendicular from the top corner straight down to the bottom side. Wherever the corner sits, that perpendicular is the same length, because the top edge and the bottom edge are a fixed distance apart. Sliding along the top does not get you any nearer the bottom. That distance has a name. It is the HEIGHT of the triangle, and it is measured straight down to the side you started from - never along a slanted side.

1:15And here is a stranger version of the same thing. Put the corner on the left-hand edge instead, and measure that triangle against the right-hand edge. Its area is ten as well. Two triangles that do not share a single side, in the same rectangle, coming out equal. So what does a triangle's area actually need you to measure? Not three sides. Just two numbers: one side, and the height belonging to that side.

1:48In this figure they are marked. Five along the bottom, four up the left. Half of five times four is ten, and ten is what we counted. Two measurements, and the rest of the triangle is not consulted at all. That was easy: the triangle came ready-boxed. Now take one that arrived on its own. Choose one of its sides to work from. At each end of that side, draw a line straight up, at right angles to it. Then draw a line through the top corner, parallel to the side you chose.

2:26Those three lines close the triangle inside a box. That box is a rectangle: two of its sides are perpendicular to the base by construction, and the fourth is parallel to it. One side of the box is the side you chose, five long. The other is the perpendicular distance up to the top corner, which is three. The box holds fifteen. The triangle holds seven and a half. Half again. But saying half again is not explaining it.

3:00So let us explain it. Drop the perpendicular from the top corner. It lands inside the base, and it cuts the box into two smaller rectangles - one of six, one of nine. The same line cuts the triangle into two pieces at the same moment. Look at the left piece. It sits in the left rectangle, and the side slanting across it runs corner to opposite corner. It is that rectangle's diagonal.

3:31A diagonal cuts a rectangle into two triangles you could lay exactly on top of each other. So the left piece is half of six, which is three. The right piece is the same story in its own rectangle. Half of nine is four and a half. Three and four and a half is seven and a half. That is where the half comes from. It is not a fudge factor bolted onto a product. It is one rectangle halved by its diagonal, used twice, and then added back up.

4:08Notice what never came into that argument. The two slanting sides of the triangle were never measured. In this one, neither of them can be: their lengths are not numbers you can write down exactly at all. It does not matter. The area is seven and a half, exactly, and nothing that was needed to get there was a slant. One side, and the perpendicular distance from it to the opposite corner. That is the whole input.

4:40One warning, because half a rectangle sounds like it should be visible in the squares. Take the rectangle five by four and its diagonal. Twenty unit squares in it. Count the squares lying wholly under the diagonal: six. Squares wholly above: six. And eight squares that the diagonal slices straight through. Six is not ten. The whole squares alone fall four short of the answer, and the eight cut ones supply exactly that four between them.

5:15So the halving is not something you read off the grid. It comes from the two halves being the same triangle laid two ways up. Now the case that looks like it should break everything. Here is a triangle whose top corner leans right out past the end of the side we want to work from. Drop the perpendicular and it lands outside the base altogether - half a base-length before the near end, off the edge of the triangle.

5:46There is no box to build: the rectangle on that side does not contain the triangle, and the perpendicular cuts nothing in two, because there is nothing on one side of it to cut. So can we still say half base times height? Run the perpendicular down and mark where it lands. Call that the foot. From the foot to the far end of the base is a distance we can box: that triangle has area eighteen.

6:17From the foot to the near end is a smaller one we can box too: area six. Take the small one away from the big one and what is left is exactly the triangle we wanted. Eighteen take away six is twelve. And half of six times four is twelve as well. The subtraction works for one reason, and it is worth saying out loud: the foot, the near end and the far end lie on one straight line in that order, so nine minus three really is the six between the two ends. Change the order and the argument goes.

6:56A triangle has three sides, so it has three of these pairs. Each side has its own height. Take this one. Side four with height three. Side three with height four. And the long side, five, whose height is two point four - not a whole number at all. Multiply each pair: twelve, twelve, twelve. Halve them: six, six, six. Which is the area - of course it is, because the triangle only has one.

7:27You may nominate whichever side you like as the base. What you may NOT do is take one side and pair it with a height belonging to a different one. That turns straight into a way of finding a length nobody handed you. Here is a triangle with a height of three onto a side of five, and another side, four long, whose height is not marked. First pair: half of five times three is seven and a half.

7:58Second pair: half of four times that unknown height is the same seven and a half. So the unknown height is fifteen over four. Three point seven five. Nobody measured it. It was forced: a triangle has one area and three ways to write it. The same trick handles the leaning triangle from before. Six times four halved is twelve, and twelve read against a side of eight gives a height of three.

8:29One last thing this buys you. Old altar builders had a problem: a shape was prescribed, and so was its area, and the two did not always agree. So they had to remake one shape as another holding the same. A rectangle six by four holds twenty-four. As a triangle, keep the six and double the height to eight. Or keep the four and double the base to twelve. Both hold twenty-four.

9:00Backwards, a triangle on six with height four holds twelve, and becomes a rectangle six by two, or three by four. Nothing keeps its sides. What is kept is the area, and the halving is what lets you trade one for the other. And finally, a comparison you can settle without a calculator. An equilateral triangle and a square, built on the same length. Which holds more? The triangle's height is one leg of a right angle whose longest side is the triangle's own side. A leg is always shorter than the longest side. So the height is less than the side.

9:43The triangle is half the side times something smaller than the side. The square is the side times the side. So the square wins, and it wins by so much that two of those triangles together still come to less than it. No square roots. Just half base times height, and knowing which length is the height.