Pythagoras' Theorem in 3D (SQA National 5 Maths): Revision Note

Using Pythagoras' theorem in 3D

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

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

• There is a 3D version of Pythagoras’ theorem:

  • is the distance between two points

  • and are the distances in the three different perpendicular directions between the two points

• A typical example of using this is to find the length of one of the diagonals that goes across the inside of a cuboid

  • In exam questions this is referred to as a space diagonal

  • You may also see people refer to it as an 'interior diagonal' or 'body diagonal'

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

  • Form two right-angle triangles

How do I apply 3D Pythagoras to more complicated problems?

• Always split up a complicated problem into 2D right-angled triangles

  • Some questions may require more than one 2D right-angled triangle

• Some 2D triangles on the diagram are still drawn in 3D

  • It helps to redraw these 2D triangles flat on the page (not at angles)

  • You can then spot any uses of Pythagoras' theorem