3D Trigonometry
Grade 7Same 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°.
