GCSE — mathsHigher2 min read

3D Trigonometry and Pythagoras

Apply Pythagoras and trigonometry to 3D problems for GCSE Maths. Find lengths and angles in cuboids, prisms, and pyramids.

Tutor AI Team2026-02-25T01:49:50.260Z

3D trigonometry extends Pythagoras and trig to three-dimensional shapes. The key skill is identifying the right-angled triangle within the 3D shape.

Core Strategy

Problem

Cuboid $4 \times 3 \times 12$ .

Face diagonal: $\sqrt{4^2 + 3^2} = 5$ .

Space diagonal: $\sqrt{5^2 + 12^2} = 13$ .

OR directly: $\sqrt{4^2 + 3^2 + 12^2} = \sqrt{169} = 13$ .

Solution

Problem

A pyramid has a square base of side 10 and height 12. Find the angle between a slant edge and the base.

Half-diagonal of base: $\frac{10\sqrt{2}}{2} = 5\sqrt{2}$ .

$\tan\theta = \frac{12}{5\sqrt{2}}$ → $\theta = \tan^{-1}\left(\frac{12}{5\sqrt{2}}\right) \approx 59.5°$ .

Solution

Problem

A line from corner A to opposite top corner G of a $6 \times 4 \times 3$ cuboid.

Base diagonal: $\sqrt{36 + 16} = \sqrt{52}$ .

Angle with base: $\tan\theta = \frac{3}{\sqrt{52}}$ → $\theta \approx 22.6°$ .

Solution

1. Find the space diagonal of a $5 \times 5 \times 5$ cube.
1. A cone has radius 4 and slant height 10. Find the angle between the slant and the base.

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Tags:

#gcse#maths#geometry#trigonometry#3d#pythagoras

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