--- title: "Pythagoras and Trigonometry in Three Dimensions" description: "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." canonical: https://lightmysky.com/learn/mathematics/pythagoras-and-trigonometry-in-three-dimensions-mt_7yKxYqQHB2 source: https://lightmysky.com/learn/mathematics/pythagoras-and-trigonometry-in-three-dimensions-mt_7yKxYqQHB2.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Geometry · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/pythagoras-and-trigonometry-in-three-dimensions-mt_7yKxYqQHB2 ## Ready when they can - Find the long diagonal of a cuboid with two uses of Pythagoras - Redraw the chosen triangle flat before calculating anything - Find the lateral edge length or the angle between an edge and the base of a right square pyramid with the apex above the base centre - 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. ## Lesson: Finding the long diagonal of a cuboid A long pencil must fit inside its box, so you need the hidden diagonal. The diagonal is hidden inside the box, so you calculate 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. You can also combine both steps into one formula: the long diagonal is the root of length squared plus width squared plus height squared. **Example.** 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. **Example.** 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. The same cuboid gives an angle too. The long diagonal meets the base at an angle, and its projection onto the base is the 5 cm base diagonal. In the upright triangle, tangent of that angle is 12 over 5, which is 2.4, so the angle is about 67.4 degrees to the nearest 0.1 degree. **Recap.** Build two flat triangles: base diagonal first, then the upright triangle that holds the long diagonal. ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Pythagoras' Theorem](https://lightmysky.com/learn/mathematics/pythagoras-theorem-mt_1VmTUxBrNd) - [Triangle Area with Half ab sin C](https://lightmysky.com/learn/mathematics/triangle-area-with-half-ab-sin-c-mt_PiwizT4D2b) ## Opens up - [Choosing a Triangle Method](https://lightmysky.com/learn/mathematics/choosing-a-triangle-method-mt_nubZzZjVvn) - [Vectors in Three Dimensions](https://lightmysky.com/learn/mathematics/vectors-in-three-dimensions-mt_o97GclnNC-)