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
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
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
There is a 3D version of the Pythagoras’ Theorem formula
d^{2} = x^{2} + y^{2} + z^{2}
However it is usually far easier to see a problem by splitting it into two or more 2D problems
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
Once you have your 2D triangle(s) you can begin to solve problems
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