The parallelogram
Push the top to the side. There is just as much room inside it.
- Base
- 16 cm
- Height
- 9 cm
- Sloping side
- 10,3 cm
- Circumference
- 52,6 cm
- Area
- 144 cm²
Regnestykket
Base × Height
16 cm × 9 cm = 144 cm²
Card: g · h
The triangle just changed places
You cut a corner off at one end and put it in at the other. Nothing was added and nothing was taken away. It is the same figure — it just lies differently. And now it is a rectangle. A rectangle is easy: you multiply the base by the height. So the parallelogram must be just as big. That is why the area is base times height.
The slope does nothing to the area
Drag the slope and watch the number for the area. It stands still. The top slides to the side, the figure becomes more and more skewed, and nothing happens to the area. That makes sense: the base is the same length all the way, and the figure is the same height all the way. There is just as much room inside it. The only thing that changes is where the top is — not how big the figure is.
The sloping side is deceptive
Look at the two numbers «Height» and «Sloping side». As soon as you push the top just a little to the side, they are no longer the same. The sloping side gets longer and longer, while the height stays put. If you multiply the base by the sloping side instead of the height, you therefore get far too big a number — and it gets worse the more skewed the figure is. That is exactly the mistake the formula is written to keep us out of.
The height is the shortest way from the base up to the top. It stands at right angles to the base, like the dotted copper line in the figure. The sloping side takes the long way — it has to go both up and to the side. That is why it is always at least as long as the height, and usually longer.
Stand it up straight
Set the slope to 0. Now the figure stands straight, and the sloping side and the height show the same number. It is the only place they agree, and it is also the only place where base times side length happens to give the right answer. A calculation that only fits at one point is not a rule. It is luck.
Pushed piles
Put a pile of cards on the table and push it askew with your hand. The pile now leans, but there are still as many cards in it, and it is still as tall. Seen from the side it has gone from being a rectangle to a parallelogram, without anything being added or taken away. It is the same observation the slider shows, just in something you can touch.
The same trick comes back
Cut the figure into pieces, rearrange the pieces into something you already know, and read off the formula. That is the whole method, and it wasn't invented just for this figure. The circle becomes a rectangle in the same way, just with many more pieces. The triangle becomes half a rectangle. The trapezium becomes a parallelogram. Once you have seen it happen, you know what to look for next time.