Area of a parallelogram, by turning it into a rectangle

0:00Here is a parallelogram. We want its area, and we are not going to invent a new formula for it. We already know a rectangle: its two sides multiplied. So the plan is not to find a formula. The plan is to find a rectangle - to cut this shape up and rebuild it as a rectangle holding exactly the same amount. The whole of the mathematics is in those three words. If they really do.

0:31First, a word about height. Pick a side to measure from. Call it the base. The height is the perpendicular distance from that side across to the one opposite. This shape has a base of fourteen, and standing on it, a height of eight. But turn it and measure from the slanted side instead. That side is ten, and its height is eleven point two. A parallelogram has two heights, not one, and they are different numbers.

1:02So it is a height, never the height. And the height is not the side next door: eleven point two is not fourteen, and eight is not ten. Now the cut. From the top-left corner, drop a perpendicular straight down onto the base. That is the height, drawn in. Cut along it. One piece is a right-angled triangle. Its two short sides are the height, eight, and the lean, six - so its slanted side is the parallelogram's own slanted side, ten.

1:40It holds twenty-four. The other piece is four-sided, with one pair of parallel sides left in it, and it holds eighty-eight. Twenty-four and eighty-eight make a hundred and twelve, the whole shape. Cutting creates and destroys nothing. Now slide the triangle across to the other end. Stop before you slide it. This is where a lesson goes wrong, because the obvious next line is: look, it fits. It does fit. But looking is not why.

2:14So ask a different question. Forget the triangle we removed. Look only at the piece that is left, and ask what shape of gap it has. What triangle would you have to drop into that end to close this into a rectangle? Build that triangle, without glancing at the one in your hand. Then compare the two. If they turn out to be the same triangle, the fit is proved. If we just slide and admire, we have taught a craft skill and no mathematics.

2:48Here is how to build it. Run the base line out past its right-hand end. From the top-right corner, drop a perpendicular down onto that extension. Where it lands, call the point Y. The triangle you have just drawn is the one that closes the gap - not chosen, constructed. And it too is right-angled, with legs of eight and six. Two right-angled triangles on the board, and we have to decide whether they are the same one.

3:22Three facts settle it. One: the new triangle's upright side is eight, the same as the removed triangle's, because both are the height of the rectangle we are building. Two: both triangles have a right angle where the perpendicular lands. Three: their slanted sides are both ten - because they are opposite sides of the parallelogram, and a parallelogram's opposite sides are equal. A right angle, a matching slanted side, and a matching upright: that is enough to force two right-angled triangles to be identical.

4:02Watch what happens if you drop the third fact. Here is a right-angled triangle whose upright is also eight. Same right angle, same eight. Its slanted side is seventeen, and it holds sixty, not twenty-four. One matching side pins down nothing at all. The fact that carries the argument is the one about the parallelogram - and it is the one students skip. So the pieces do fit, and we have a rectangle holding a hundred and twelve.

4:38Its short side is the height, eight. Its long side is - and here is the second place to be careful - not obviously the base. It looks like the base. That is not a reason. Here is the reason. The off-cut at the left end is six. The off-cut at the right end is six as well, because it is the base of the triangle we just proved congruent.

5:04Between them lies a stretch of eight, and that stretch belongs to the base and to the rectangle both. The base is that eight plus the left six. The rectangle's long side is that same eight plus the right six. Both are fourteen. One shared stretch, added to two equal off-cuts. That is why the widths match. Fourteen times eight is a hundred and twelve. Area equals base times height, and it is the rectangle formula in a disguise.

5:40Cutting a shape into pieces and rebuilding them into a different shape of the same size has a name: dissection. It is very old, and the altar geometry of ancient India is full of it. And nothing in what we did depended on which side we started from. Take the slanted side as the base instead. It is ten. Drop a perpendicular across to the side opposite: eleven point two.

6:09Ten times eleven point two is a hundred and twelve. The same shape, the same amount, measured completely differently. Any side will do, as long as you use the height that belongs to it. One honest warning, because anyone who draws a very slanted example will hit it. Lean the shape far enough and the perpendicular from the top corner lands past the end of the base, outside the shape altogether.

6:41Then the triangle the argument talks about does not exist, and the cut cannot be made as described. Nothing is broken. The shape still holds a hundred and eighty. Use the other pair of sides. That side is twenty-five, its height is seven point two, and twenty-five times seven point two is a hundred and eighty. The escape was built in from the start: either pair may be the base.

7:12Now watch what the formula does not care about. Here are seven parallelograms. Every one stands on a base of five, and every one is three tall. They lean by different amounts, and not even all the same way - three lean right, four lean left. Every single one holds fifteen. Base times height cannot see the lean. It never asked. The distance round the outside is a different story. The slanted side grows with the size of the lean, whichever way it goes.

7:46The smallest lean here is about a third of a square, and the largest is nearly three. So the first one has the shortest way round and the last one the longest, and no two of the seven agree. One area, seven perimeters. Four to try. Each time, a side and the height that belongs to it. Seven and four gives twenty-eight. Five and three gives fifteen. The next two are measured from a slanted side, not a level one. Five and four point eight gives twenty-four; two and four point four gives eight point eight.

8:27Look at the last pair: the height is four point four, the side only two. A height can be longer than its own side - it is a distance across the shape, not an edge of it. And one run backwards. A parallelogram with twelve along the bottom, standing six tall, holds seventy-two. Its other side is seven point six. So seven point six times the missing height is seventy-two, and the height is a hundred and eighty nineteenths - a shade under nine and a half.

9:02That is bigger than six. The shorter side always carries the taller height, because the two products have to come out the same. One last comparison, and it decides whether any of this was understood. A rectangle five by four. A parallelogram whose sides are also five and four. Same two sidelengths. Both have a distance of eighteen round the outside. Stand them on the same base of five. For the rectangle, the four IS the height - it stands straight up.

9:37For the leaning one, the four is the slanted side, and the height is a leg of the right triangle underneath it. Here it is three point two. Twenty against sixteen. And that is not special to this lean. Any lean at all makes the height shorter than the side, because a slanted side has to cover sideways distance as well as upward distance. So the same two sides do not mean the same area. What fixes the area is a base and its height - and that is exactly what the rectangle we built was telling us all along.