Angles between lines and planes

The angle between a line and a plane is the acute angle between the line and its perpendicular projection onto that plane. In a cuboid, a space diagonal projects onto a base diagonal. The triangle formed by the space diagonal, the base diagonal and the height is right-angled, so the tangent of the required angle is height divided by base-diagonal length.

For the angle between two intersecting planes, take a cross-section perpendicular to their line of intersection. The angle between the two cross-section lines gives the dihedral angle. A triangle in an arbitrary cross-section may give the wrong angle. Non-right triangles in a solid can require the sine or cosine rule after the correct lengths have been found.

Worked example

A cuboid has base edges 6 cm and 8 cm and height 5 cm. Find the angle its space diagonal makes with the base.

1. The projection onto the base has length √(6² + 8²) = 10 cm.
2. The perpendicular height is 5 cm, so tan θ = 5/10.
3. θ = arctan(1/2), using degree mode.

Answer: Approximately 26.6°

Revise first: Pythagoras in three dimensions, Sine rule, cosine rule and triangle area.