Area of a trapezium, derived two different ways

0:00Here is a trapezium. One pair of its sides is parallel and the other is not. The parallel ones are 6 and 27, and they are 12 apart. Notice what is not true here. Its slanted sides are 13 and 20, so it is not symmetric - and neither of them is the height. The height is not a side at all: it is the distance across, from one parallel side to the other.

0:30We are not going to reach for a formula. We are going to cut this into things we already handle. Drop a perpendicular from each end of the short side down to the long one. A triangle, a rectangle, and another triangle. One thing has to be checked first, because the whole calculation rests on it. Is that middle piece really a rectangle? Two of its corners are right angles by construction - we drew those lines perpendicular.

1:02The other two have to be argued. They are right angles because the top side is parallel to the bottom one: each perpendicular cuts across both parallels, and the two angles it makes on the same side of it total a straight angle. One is a right angle, so the other one is too. Four right angles. It is a rectangle: both horizontal sides 6, both uprights 12. And the two end pieces are right-angled triangles, square at the foot of each perpendicular and nowhere else.

1:41Each triangle stands on a stub of the bottom side, and those stubs are what we do not know. Call them x and y - here, 5 and 16. The first triangle is half of 5 times 12, which is 30. The rectangle is 6 times 12, which is 72. The last triangle is half of 16 times 12, which is 96. Thirty, seventy-two and ninety-six - together, 198, which is exactly what the four corners of the whole shape say it holds.

2:18But an answer that needs x and y is not much of an answer, because nobody hands you those. Write the three areas with the letters left in. Half h x, plus h a, plus half h y. Every term has an h, so pull it out: what is left inside is half of x, plus a, plus half of y. And here is the whole trick. The x and the y never appear apart. Only their total does.

2:49And their total is sitting on the bottom side: the bottom is the top plus the two stubs, so the stubs together are the bottom minus the top. Here that is 5 plus 16, which is 21, which is 27 minus 6. You never measure either one. That is why this shape needs three measurements, not five. Put the total back in: half h, times b minus a, plus 2 a. The minus a and one of the two a's cancel.

3:23Half the height, times the sum of the two parallel sides. Twelve halved is 6. Six plus twenty-seven is thirty-three. And six times thirty-three is one hundred and ninety-eight - the answer the pieces gave. Now, the half. It is the most-dropped symbol in this subject. Leave it out and you get three hundred and ninety-six. That is not slightly wrong. It is exactly twice the shape, every time. So let us make the half impossible to forget.

3:59Half of the sum of the two parallel sides is 16.5. Do not read that as half of an area. Read it as a length. It is a length you can point to. Go halfway up and draw a line straight across, from the middle of one slanted side to the middle of the other. That segment is 16.5 long, and parallel to both. Longer than the short side, shorter than the long one. It is the average width - the width this shape would have if it did not taper.

4:33So the formula is not a formula at all. It is the rectangle rule: width times height. Sixteen point five by twelve. And you can build that rectangle. Swing the triangle below the halfway mark on the left outward, and it lands exactly on the gap above it. The same on the right - and those two are not the same triangle, so both swings have to be checked. Average width times height. Nobody reading it that way loses the half.

5:09Now a shape that breaks the picture we have been drawing. The top side is pushed so far over that the perpendicular from its left end lands outside the bottom side altogether. The height has to be drawn out in space, beside the shape rather than inside it. It is still a trapezium: parallel sides 6 and 16, and 8 apart. Does the formula still hold? Half of 8 is 4, times 22, is 88. And the four corners say 88.

5:44It holds. But the picture that produced it has fallen apart, so we owe ourselves a reason. There are two, and both are worth having. The first keeps the three pieces and changes their signs. Drop both perpendiculars anyway and take the rectangle between the two feet: 6 by 8, holding 48. At the left end the triangle between rectangle and shape lies outside the shape, so it comes off: 20. At the right end it lies inside, so it goes on: 60.

6:21Forty-eight, less twenty, plus sixty. Eighty-eight. And the two triangles are not equal, so those signs are doing real work. On the ordinary trapezium the very same three pieces are all added, because both feet land on the bottom side. That is the whole difference: a sign flipped. Not the formula. The second draws one line: from the top right corner, parallel to the left slanted side, down to the bottom.

6:55What it cuts off is a parallelogram - top and bottom already parallel, and the two sides parallel because we drew one that way. So it holds base times height: 6 times 8, which is 48. What is left is a triangle standing on the difference of the two parallel sides, 16 minus 6, which is 10. Half of 10 times 8 is 40. Forty-eight and forty. Eighty-eight again, from a different cut.

7:28One warning: if the two parallel sides are equal, the shape is already a parallelogram and there is no triangle left behind. The approach is fine, but that case needs checking. There is a third way, and it explains the half rather than producing it. Take a second copy. Turn it through a straight angle about the middle of one slanted side, and slide it against the original. What you get is a parallelogram.

8:01Its base is the two parallel sides laid end to end: 6 and 27, so 33. Its height is 12. It holds 396. And the trapezium is half of it: one hundred and ninety-eight. There is the half, and it is not algebra any more. You built two trapeziums' worth of area on purpose, and gave half of it back. But why is that a parallelogram? Why is it even four-sided?

8:33Join two copies of a shape along a side and what you normally get is six-sided, with a corner sticking out at each end of the join. Watch a quadrilateral with no parallel sides. Turn a copy, join it, and the area doubles just the same - but the corners stay corners. Six of them. The trapezium's straighten out instead. At each end of the join two angles meet, and they are the two that sit on the same side of a line crossing the parallel pair. They total a straight angle, so the two edges become one.

9:13That is where the parallel sides get used, and it is the only place they are used. Without it, the parallelogram is just something the picture looked like. Four to try. Two are drawn turned, which changes nothing - the parallel sides are whichever ones are parallel, and they need not be horizontal. Ten and seven, 16 apart: 136. Twenty-four and thirty-six, 14 apart: 420. Fourteen and six, 10 apart: 100. And twelve and eighteen, 8 apart: 120.

9:55It runs backwards too. Given 420, a height of 14 and one side of 24, the other is 36. And now the last thing, worth more than the derivation. Set the two parallel sides equal - both 9 - and it gives 9 times the height. That is the parallelogram rule. Let one shrink away to nothing, and it gives half the base times the height. That is the triangle rule.

10:25So this is not a fourth thing to memorise. It is the one that contains the other two.