The sloping side
Four identical triangles, moved around — and suddenly you can work out the sloping side.
- The base
- 6 cm
- The height
- 4 cm
- The base times itself
- 36 cm²
- The height times itself
- 16 cm²
- The two added together
- 52 cm²
- The sloping side
- 7,2 cm
Regnestykket
√(Base × Base + Height × Height)
√(6 cm × 6 cm + 4 cm × 4 cm) = 7,2 cm
Card: √(a² + b²)
Nothing disappeared
The big square stays the same the whole way through — drag the slider and watch the edge, it doesn't move. Inside it are four triangles, and they are the same four too: none appears, none disappears, none gets bigger. The only thing that happens is that they are pushed to other places. So the empty space left over must be the same size before and after. That is the whole proof, and that is why there is no calculation in the figure — you can see it.
Two empty squares become one
Before you unfold, there are two empty squares left. One has the base as its side, the other has the height. After you have unfolded, there is one empty area left, and it stands at an angle — but it is still a square with four equal sides, and those sides are exactly the sloping side. The numbers in the figure say the same: the two numbers you could read before are added together into the one shown afterwards. Try setting the base to 6 and the height to 4: it shows 36 and 16, and afterwards it shows 52.
Why the sloping side rarely comes out even
The base and the height are whole centimetres, so the two square numbers are also whole numbers. But the sloping side is the square root of the sum, and it almost never hits a whole number. With 6 and 4 it comes to 7.2 — not 7 and not 8. That isn't an inaccuracy in the drawing; it's how the numbers are. There are a few pairs that come out even, and they are worth looking for: try 3 and 4, and try 6 and 8. Tradespeople have used exactly those for thousands of years to make a corner stand perfectly square.
Set the two sliders to the same
Then the two empty squares become the same size, and the sloping side divides the big square exactly in half. It is the only setting where the figure is symmetrical, and the sloping side becomes the base multiplied by the square root of two — about one and a half times as long. You meet that number again every time something has to be cut on the diagonal, and it is worth noticing that it doesn't depend on how big the square is. Move both sliders up and down together: the ratio between the sides stays the same every time.
The roof is the same triangle
Raise a roof over a house. The base is half the house's width, measured from the outer wall in to the middle. The height is how high the ridge rises above the ceiling. And the sloping side is the piece of timber that runs from the wall plate up to the ridge — the rafter. The carpenter can measure the width and decide the height, but he can't measure the length of the timber on the house, because the house isn't standing yet. He works it out, with exactly the calculation shown here.
And the same rule turns up everywhere
A ladder leaning against a wall. A staircase, where the steps are the base and the height and the handrail is the sloping side. A pipe to be run at an angle through a room. A cable tray to be bent round a corner. Each time you know two sides at right angles to each other, and each time you need the third. You don't need to remember a name for the rule to use it — you just need to see that it is the same figure again.