CIE IGCSE Maths

Revision Notes

6.5.1 3D Pythagoras & SOHCAHTOA

What are Pythagoras & SOHCAHTOA?

  • Pythagoras’ Theorem helps us find missing side lengths of a right-angled triangle
  • It is also frequently used for finding the distance (or length) of a line

3DPythagTrig Notes fig1 (1), downloadable IGCSE & GCSE Maths revision notes

3DPythagTrig Notes fig1, downloadable IGCSE & GCSE Maths revision notes

  • SOHCAHTOA is an acronym for the three trigonometric ratios that connect angles (θ) and sides (Opposite, Hypotenuse and Adjacent) in a right-angled triangle
    • Sine – SOH – sin θ = O ÷ H
    • Cosine – CAH – cos θ = A ÷ H
    • Tangent – TOA – tan θ = O ÷ A

3DPythagTrig Notes fig2 (1), downloadable IGCSE & GCSE Maths revision notes

3DPythagTrig Notes fig2, downloadable IGCSE & GCSE Maths revision notes

How does Pythagoras work in 3D?

  • 3D shapes can often be broken down into several 2D shapes
    • For example nets and surface area
  • With Pythagoras’ Theorem problems you will be specifically looking for right‑angled triangles
    • The right-angled triangles you need will have two known sides and one unknown side

3DPythagTrig Notes fig3 (1), downloadable IGCSE & GCSE Maths revision notes 3DPythagTrig Notes fig3 (2), downloadable IGCSE & GCSE Maths revision notes 3DPythagTrig Notes fig3 (3), downloadable IGCSE & GCSE Maths revision notes

  • There is a 3D version of the Pythagoras’ Theorem formula
    • d2 = x2 + y2 + z2
  • However it is usually far easier to see a problem by splitting it into two or more 2D problems

3DPythagTrig Notes fig4, downloadable IGCSE & GCSE Maths revision notes

How does SOHCAHTOA work in 3D?

  • Again look for a combination of right-angled triangles that would lead to the missing angle or side
  • The angle you are working with can be awkward in 3D
    • The angle between a line and a plane is not obvious
    • If unsure, put a point on the line and draw a new line to the plane
      This should create a right-angled triangle

3DPythagTrig Notes fig5, downloadable IGCSE & GCSE Maths revision notes

3DPythagTrig Notes fig6, downloadable IGCSE & GCSE Maths revision notes

  • Once you have your 2D triangle(s) you can begin to solve problems

3DPythagTrig Notes fig7 (1), downloadable IGCSE & GCSE Maths revision notes 3DPythagTrig Notes fig7 (2), downloadable IGCSE & GCSE Maths revision notes 3DPythagTrig Notes fig7 (3), downloadable IGCSE & GCSE Maths revision notes

Exam Tip

Add lines/triangles/etc. to any given diagram to help you see the problem and draw any 2D triangles separately as a 3D diagram can get hard to follow.

Worked Example

3DPythagTrig Example qu, downloadable IGCSE & GCSE Maths revision notes

3DPythagTrig Example sol (1), downloadable IGCSE & GCSE Maths revision notes 3DPythagTrig Example sol (2), downloadable IGCSE & GCSE Maths revision notes

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