3D Pythagoras and Trigonometry
This free Higher GCSE Maths worksheet on 3d Pythagoras and trigonometry helps you revise using Pythagoras’ theorem to find missing sides. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. Find a right-angled triangle inside the solid, then use the 2D rules.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
Three-dimensional Pythagoras and trigonometry extend the two-dimensional methods into solids, most often cuboids.
The key technique is to find a right-angled triangle inside the solid and work with that. Usually this means finding a diagonal across the base first, then using it as one side of a second triangle that reaches the top corner.
For a cuboid, the space diagonal can be found in one step with \(d = \sqrt{l^2 + w^2 + h^2}\), which is simply Pythagoras applied twice. Sketching the triangle you are using, separately from the solid, makes these questions far more manageable.
Revision notes
Working in two stages
Find the diagonal of the base using Pythagoras, then use that as one side of a vertical triangle.
For a cuboid \(3 \times 4 \times 12\): the base diagonal is \(\sqrt{9+16} = 5\), then the space diagonal is \(\sqrt{25 + 144} = 13\).
The direct formula
For a cuboid, \(d = \sqrt{l^2 + w^2 + h^2}\) gives the space diagonal in one step.
This is just the two-stage method combined, and either approach earns full marks provided the working is clear.
Angles in three dimensions
To find an angle between an edge and the base, use the right-angled triangle containing both.
The base diagonal is usually the adjacent side and the vertical edge the opposite, so tangent is the ratio needed.
Key points
- Find a right-angled triangle inside the solid.
- Work in two stages: base diagonal, then vertical triangle.
- Cuboid space diagonal is \(\sqrt{l^2 + w^2 + h^2}\).
- Sketch the triangle separately from the solid.
- Use tangent for angles between an edge and the base.
- Keep full accuracy between stages.
Worked examples
Example 1
Find the space diagonal of a cuboid \(3 \times 4 \times 12\)cm.
Working
Example 2
Find the space diagonal of a cube of side \(6\)cm to 1 d.p.
Working
Example 3
A cuboid has base diagonal \(8\)cm and height \(6\)cm. Find the angle between the space diagonal and the base.
Working
Common mistakes
Trying to work in three dimensions at once.
Break the problem into two right-angled triangles and handle each separately.
Using an edge instead of the base diagonal.
The vertical triangle uses the base diagonal, not one of the base edges.
Rounding the base diagonal before the second stage.
Keep full accuracy or the final answer will be slightly wrong.
Forgetting the square root.
Pythagoras gives the square of the length, so the root is the final step.
Exam tips
- Sketch the right-angled triangle separately from the solid.
- Label which length is which before calculating.
- Carry full accuracy between the two stages.
- Use the direct formula for a cuboid space diagonal if you prefer.
Key terms
- Space diagonal
- The longest diagonal, corner to opposite corner through the solid.
- Base diagonal
- The diagonal across the bottom face.
- Cuboid
- A box-shaped solid with six rectangular faces.
- Plane
- A flat surface, such as the base of a solid.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.