The triangle
Push the top far out to the side. There is still just as much room in it.
- The base
- 10 cm
- The height
- 7 cm
- Area
- 35,0 cm²
- Left side
- 8,6 cm
- Right side
- 8,6 cm
- All the way round
- 27,2 cm
Regnestykket
Base × Height ÷ 2
10 cm × 7 cm ÷ 2 = 35,0 cm²
Card: b · h ÷ 2
Two identical triangles became one box
What appeared from behind is not a new triangle. It is the same one, copied and turned half a turn round. Notice that it lands right up against the first one: there is no gap between them, and no corner sticking out over the other. Together the two triangles cover a slanted box, and they split it exactly in two. So one triangle is exactly half of that box. The bottom is called the triangle's base, and the vertical distance up to the top is called the height. The box has the same base and the same height as the triangle you started with.
The slanted box is just as big as an ordinary one
A slanted box looks as if it's hard to measure, but it isn't. Imagine cutting the sloping end off on the left and moving the piece over to the right. It fits exactly, because the two sloping edges lean the same way. Now there is an ordinary rectangle with the same base and the same height, and it takes up base times height. So the slanted box also takes up base times height. And the triangle is half of it: base times height, divided by two.
The top can go wherever it likes
Drag the last slider and watch the area. The top slides far to the left and far to the right, the triangle becomes pointed and skewed and leans out over its own base — and the number for the area doesn't move once. As long as the base stays the same and the top stays at the same height, there is just as much room in the triangle. Think of a stack of paper lying on the table. If you push the top of the stack to the side so it leans, there are still exactly as many sheets in it. No paper has been added, and nothing has disappeared. The triangle behaves in the same way.
The sides change, the area doesn't
Look at the two sloping sides and the number for all the way round while you push the top. Those numbers go up and down all the time. It is worth stopping to think about, because it is easy to believe a figure gets bigger when the edges get longer. It doesn't. The length all the way round and the room inside are two different things, and one says nothing certain about the other.
The height must stand at right angles
The dotted line from the top down is the height, and it always stands at right angles to the base. It isn't the same as one of the sloping sides, even though they can look alike when the triangle is narrow. If you push the top right outside the base, something happens that looks wrong but is right: the line lands beside the triangle, on a thin line that extends the base. The height is measured to the line the base lies on, not just to the part of it you can see. The calculation is the same.
Where you'll meet it again
The gable of a house is a triangle. So is a roof surface seen from the end, a triangular offcut of a sheet and the gap between two sloping pipes. Every time someone needs to know how much to use, or how much is left, it is the same calculation: measure the base, measure the height at right angles to it, multiply them and divide by two. And you don't need to think about how pointed the triangle is.