Three-dimensional problems

Last time

Before we start: last time, in Area of a triangle, sine and cosine rules, 3D problems, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

The longest stick in the box

How long a stick fits inside a cardboard box? Not along the bottom edge. The longest stick goes from one corner to the opposite corner, through the air. To find it, you do two flat right-angled triangles, one after the other. Every three-dimensional problem comes down to spotting right-angled triangles in the solid.

The golden rule

A solid is hard to see, but each triangle inside it is flat. Pick out the triangle that contains the length or angle you want. Redraw that triangle on its own, with the lengths you know, and the right angle marked. Then use Pythagoras or trigonometry exactly as you would in two dimensions.

Diagonal of a cuboid

A cuboid is eight long, six wide and five high. First the diagonal of the base: the root of eight squared plus six squared, which is ten. Now a vertical triangle with the base diagonal and the height. The root of a hundred plus twenty-five is eleven point one eight. In one go: the space diagonal is the root of length squared, plus width squared, plus height squared.