The trapezium
An identical one, turned round, fits exactly alongside.
- Top side
- 6 cm
- Bottom side
- 10 cm
- Height
- 5 cm
- Average of the sides
- 8,0 cm
- Area
- 40,0 cm²
Regnestykket
Average of the sides × Height
8,0 cm × 5 cm = 40,0 cm²
Card: (a + b) ÷ 2 · h
Two identical ones become a parallelogram
You took an identical one, turned it half a turn and put it alongside. The two fit together without a gap and without overlapping, and that is no coincidence: the sloping side you joined them along is the same side on both figures. One just faces the other way. Together they are now a parallelogram. The base of that parallelogram is the top side plus the bottom side laid end to end, and the height is the same as before.
A parallelogram is base times height. Here the base is the two sides added together, so the two trapeziums together take up the two sides times the height. One trapezium is half of that. So: add the two parallel sides, multiply by the height, and halve. That is the whole formula. You can also see the halving directly: the short copper line across the middle is exactly half as long as the finished parallelogram is wide.
The average is a real length
Halving the sum of the two sides is the same as taking their average. And that average isn't just a number in a calculation — it is the copper line across the middle of the figure, the one marked «average». Measure it, and you get exactly the number shown next to «Average of the sides». The area of the trapezium is that line times the height. So the trapezium is as big as a rectangle that is as tall and as wide as the middle line.
Turn the top side down to 0
The top disappears, and the figure becomes a triangle. The formula does nothing special about it: it still adds the two sides, but one of them is zero. Half the bottom side times the height. That is the triangle's formula, and it isn't a new rule you have to learn on the side. It is the same rule with a zero in it.
Make the two sides equal
Now the figure is a parallelogram — the same slanted parallelogram as in the other workshop. The formula adds the two equal sides, halves again, and is back at the one side. Base times height. Here too nothing new happens; the calculation just collapses into the short one.
It is the wide one of them
The trapezium isn't one more figure to keep track of. It is the general one, and the triangle and the parallelogram are two special cases of it, which you can reach with the top slider. If you can remember the trapezium's formula, you can work out the other two from it without knowing them by heart.
Where you'll meet it
The trapezium turns up every time something slopes in towards the top. Cut straight across a ditch or a pile of gravel with sloping sides, and the cross-section is a trapezium. The same goes for the cross-section of many gutters and channels. If you need to know how much is in it, it is the trapezium's area you calculate — the two parallel sides and the distance between them.