Pythagoras and Trigonometry in Three Dimensions
Find lengths and angles inside cuboids, prisms and pyramids by picking a right-angled triangle out of the solid and solving it on its own.
What a learner can do afterwards
- Find the long diagonal of a cuboid with two uses of Pythagoras
- Redraw the chosen triangle flat before calculating anything
- Find the angle between an edge and the base of a pyramid
- Find the slant height of a right square pyramid with the apex directly above the base centre, from half the base and the vertical height.
1 · Read
A long pencil must fit inside its box, so you need the hidden diagonal. No ruler can reach through the box, so you build it in two steps. First find the diagonal across the base. Then use that diagonal with the height to make a second right-angled triangle.
Take a cuboid 3 cm by 4 cm by 12 cm. Its base diagonal is 5 cm, since 3 squared plus 4 squared is 9 plus 16, which is 25. Then the long diagonal is 13 cm, since 5 squared plus 12 squared is 25 plus 144, which is 169.
Redraw each triangle flat before you touch your calculator. Mark the right angle and label the hypotenuse. Then square back at the end: the two shorter squares must add to the longest square.
For a right square pyramid, with the apex directly above the base centre, the angle between an edge and the base sits in a skinny upright triangle. The height is one side and the line from the base centre to a corner is the other. With height 8 cm and centre to corner 6 cm, the edge is 10 cm, since 64 plus 36 is 100. In this centred pyramid, the face slant height sits in a different upright triangle, with half the base edge and the vertical height as its two short sides.
Build two flat triangles: base diagonal first, then the upright triangle that holds the long diagonal.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.