3D Trigonometry

Grade 7

Same trig, new dimensions

SOHCAHTOA still applies in 3D problems — you just need to identify the correct right-angled triangle in the 3D shape. The key skill is extracting a 2D triangle from the 3D shape and working in that plane.

Common angles to find: the angle of elevation (looking up from horizontal), the angle between a line and a base plane, or the angle inside a pyramid between a slant edge and the base.

Strategy

1. Sketch the 3D shape.

2. Identify the right-angled triangle that contains the angle you want.

3. Find the missing side(s) using Pythagoras if needed.

4. Apply SOHCAHTOA to find the angle.

The base of the right-angled triangle is often a diagonal of a face, not an edge of the solid. Finding that base length with Pythagoras first is the classic two-step approach.

Worked example

EXAMPLE

A rectangular box has base 6 cm × 8 cm and height 10 cm. Find the angle between the space diagonal and the base.

Base diagonal = √(6² + 8²) = √100 = 10 cm. The space diagonal, the base diagonal, and the vertical height form a right-angled triangle. tan θ = height ÷ base diagonal = 10 ÷ 10 = 1. θ = tan⁻¹(1) = 45°.

Try this one

A cuboid box has a base measuring \( 8 \) cm by \( 6 \) cm and a height of \( 5 \) cm. A drinking straw is placed inside so that it runs from one bottom corner to the top corner diagonally opposite.

Calculate the angle that the straw makes with the base of the box. Give your answer to \( 1 \) decimal place. [4]

Nova's hint:

You need the base diagonal first, then trigonometry in an upright triangle.

Ask yourself: 'which length is opposite the angle and which is adjacent to it?' The height rises above the base diagonal, so once both are known you can use the tangent ratio.

Try to Solve it with Nova?