20.5 3D Pythagoras and trigonometry — Learn

Prerequisites: Pythagoras' theorem and trigonometric ratios in a flat triangle.

Objectives: find a diagonal or angle in a three-dimensional shape by identifying right-angled triangles.

A 3D diagonal joins two vertices that do not lie on one face. Work in stages: find a face diagonal first, then use it with the remaining perpendicular side.

Example 1. A cuboid measures $3$ cm by $4$ cm by $12$ cm. Find its space diagonal $d$.

First find the diagonal across the $3$ by $4$ base.

Now use $b$ and the height as the two perpendicular sides.

Example 2. In the same cuboid, find the angle between the space diagonal and the base. The diagonal's vertical rise is $12$ cm and its horizontal projection is $5$ cm.

Watch out. A face diagonal is not the space diagonal. Draw the second right-angled triangle and use the face diagonal as one of its sides.

Progress check

The garden diagonal in 20.1 is a flat version of the same method: identify perpendicular sides before applying the theorem.

You should now be able to

• identify the face diagonal and use it to find a space diagonal;
• model the angle between a space diagonal and a face using a right-angled triangle;
• select Pythagoras' theorem or a trigonometric ratio and check the units.