SQA National 5 Mathematics

3D Pythagoras

Using right-angled triangles inside 3D shapes.

Formula reference

Check the National 5 rules and formulae linked to this topic.

Match exam clues to a suitable method.

National 5 Mathematics lesson

3D Pythagoras usually needs two steps. First find a diagonal on a rectangular face, then use that diagonal in a second right-angled triangle.

In a cuboid with length l, width w and height h, the space diagonal d satisfies d² = l² + w² + h².

A clear sketch is important because the right angle may be hidden inside the 3D shape.

Two Pythagoras steps inside a cuboid

Using only one face when the question asks for the internal diagonal.

Compare your answer with the size you expected from the question.

If a 3D question feels difficult, draw the base diagonal first and label it before doing the second triangle.

Use brackets for fractions, powers and square roots, then round only at the final line.

Worked example 1

A cuboid is 6 cm by 8 cm by 10 cm. Find the space diagonal.

Answer: The space diagonal is 10√2 cm, about 14.1 cm.

Worked example 2

A box measures 3 m by 4 m by 12 m. Find the longest internal diagonal.

So: The longest internal diagonal is 13 m.

Worked example 3

A cuboid has length 9 cm, width 12 cm and space diagonal 17 cm. Find its height.

Answer: The height is 8 cm.