3D Pythagoras & trigonometry – GCSE Maths

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3D Pythagoras and trig are 2D in disguise. Draw the right-angled triangle that actually contains the length or angle you want. A cuboid's space diagonal is $\sqrt{l^2+w^2+h^2}$ (face diagonal first, then a second Pythagoras). An angle between a line and the plane is the angle between the line and its shadow on the plane.

SOHCAHTOA on that slice: the height of a pyramid, the diagonal of a room, the angle a space diagonal makes with the floor. Never apply a trig ratio to a 3D sketch without a right-angled 2D triangle drawn and labelled. Keep unrounded lengths for a follow-up angle.

Space diagonal of a cuboid $\sqrt{l^2+w^2+h^2}$. Often two steps: face diagonal, then space diagonal. An angle with the base: $\tan \theta =\frac{\text{height}}{\text{base diagonal}}$.

Objectives

• Calculate the face diagonal of a cuboid using 2D Pythagoras.
• Calculate the space diagonal using 3D Pythagoras.
• Find tan and sin of the angle between the space diagonal and the base.

Key points

• Find the diagonal across a face using 2D Pythagoras.
• Find the space diagonal of a cuboid using $d^2=a^2+b^2+c^2$.
• Identify the angle between a line and a plane and the right-angled triangle that contains it.
• Use SOH CAH TOA in that triangle to find the angle.

Worked examples

Questions and answers

GCSE Maths 3d pythagoras & trigonometry questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Stopping at the face diagonal

   The space diagonal passes through the 3D interior and must include all three dimensions

2. Using the space diagonal in tan

   tan θ = opposite / adjacent uses the base diagonal, not the space diagonal

3. Leaving the fraction unsimplified

   Always cancel fractions to simplest form (e.g. 24/10 = 12/5)

Exam tips

• A calculator is useful here. Show the calculation you typed, not only the final line.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• Always redraw the 2D right-angled triangle next to the 3D diagram; it earns a method mark and stops mistakes.
• Write which trig ratio you are using and why ('opposite and adjacent, so tan').

Printable worksheet

The same questions are in a PDF you can print. Answers start on a later page.

Download worksheet (PDF)

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More on 3D Pythagoras & trigonometry

• 3D Pythagoras & trigonometry worked examples and method
  The method in order, key facts and the full lesson explanation
• 3D Pythagoras & trigonometry practice questions
  A second set of questions with answers, plus quick checks

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.