IGCSEMathsEdexcelMaths BRevision NotesTrigonometry3D Pythagoras & Trigonometry3D Pythagoras & Trigonometry

3D Pythagoras & Trigonometry (Edexcel IGCSE Maths B): Revision Note

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Amber

Written by: Amber

Reviewed by: Jamie Wood

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Edexcel Maths BEdexcel Maths A (Modular): Higher Unit 1Edexcel Maths A: HigherCambridge (CIE) International Maths: ExtendedCambridge (CIE) Maths: Extended

3D Pythagoras & trigonometry

How do I use Pythagoras' theorem in a 3D shape?

  • You can often find right-angled triangles within 3D shapes

    • If two sides of the triangle are known, you can use Pythagoras’ theorem

Example showing Pythagoras' theorem used to find the slant height of a cone given its radius and perpendicular height.

Is there a 3D version of the Pythagoras' theorem formula?

  • There is a 3D version of Pythagoras’ theorem: d squared equals x squared plus y squared plus z squared

    • d is the distance between two points

    • x comma space y and z are the distances in the three different perpendicular directions between the two points

Example showing Pythagoras' theorem to find the diagonal in a cuboid (3D formula).

  • However, all 3D situations can be broken into two 2D problems

    • Form two right-angle triangles

Example showing Pythagoras' theorem to find the diagonal in a cuboid (3D formula).
3DPythagTrig Notes fig3 (3), downloadable IGCSE & GCSE Maths revision notes

Examiner Tips and Tricks

You are not given the 3D Pythagoras formula in the exam.

You can always split 3D problems into two 2D problems (which don't need this formula).

How do I use SOHCAHTOA in 3D?

  • Again, look for right-angled triangles to use with SOHCAHTOA

    • You may need combinations of triangles that lead to the missing side or angle

Example showing SOHCAHTOA to find the angle between the base and the slant height of a cone , given its radius and perpendicular height.

Examiner Tips and Tricks

If you are stuck in the exam with a complicated 3D diagram, it is always better to just start finding any lengths and angles in the shape, as:

  • these may end up being useful

  • you may score more marks than if you had left the question blank

Worked Example

A pencil is being put into a cuboid shaped box.

The box has dimensions 3 cm by 4 cm by 6 cm.

A cuboid ABCDEFGH. AB = 3 cm, AD = 4 cm, DG = 6 cm.

(a) Find the length of the longest pencil that can fit inside the box.

Answer:

The longest possible pencil will fit between diagonally opposite vertices, e.g. AF

Form a 2D right-angled triangle, such as triangle ABF

Cuboid ABCDEFGH with a right-angled triangle ABF highlighted and another right-angled triangle BEF also highlighted.

Method 1

To find the length AF, there are a few different options
One option is to find length BF (from triangle BEF) then AF (from triangle ABF)
Draw triangle BEF flat and use Pythagoras' theorem to find BF

Triangle BEF, BE = 6 cm, EF = 4 cm, BF = a cm.

a squared equals 4 squared plus 6 squared a squared equals 16 plus 36 a squared equals 52

Draw triangle ABF flat and use Pythagoras' theorem to calculate AF

Triangle ABF. AB = 3 cm, BF = a cm and AF = b cm.

table row cell b squared end cell equals cell 3 squared plus a squared end cell row cell b squared end cell equals cell 9 plus 52 end cell row cell b squared end cell equals 61 row b equals cell square root of 61 equals 7.81024... end cell end table

The longest pencil that can fit inside the box is 7.81 cm (3 s.f.)

Method 2

Apply the 3D version of Pythagoras’ theorem: d squared equals x squared plus y squared plus z squared

The distance in the x direction is 4 cm
The distance in the y direction is 6 cm
The distance in the z direction is 3 cm

table row cell d squared end cell equals cell 4 squared plus 6 squared plus 3 squared end cell row d equals cell square root of 4 squared plus 6 squared plus 3 squared end root end cell row d equals cell square root of 61 equals 7.81024... end cell end table

The longest pencil that can fit inside the box is 7.81 cm (3 s.f.)

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Amber

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

Jamie Wood

Reviewer: Jamie Wood

Expertise: Maths Content Creator

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.