Angles between two planes

Two intersecting planes share a line. To measure their angle, choose one line in each plane perpendicular to that shared line, meeting at the same point. The angle between those perpendicular lines gives the plane angle. An arbitrary diagonal on a face generally gives a different angle. Identify the cross-section before selecting a trigonometric ratio.

In a right pyramid with apex vertically above the centre of a rectangular base, a section through the apex and perpendicular to a base edge meets that edge at its midpoint. The base-centre to edge distance is half the corresponding width. The apex height is perpendicular to the base, so the section is right-angled. Use exact lengths until the final angle calculation.

Worked example

A right square pyramid has base side 12 cm and perpendicular height 8 cm. Find the acute angle between a triangular face and the base.

1. Take the section through the apex, base centre and midpoint of an edge. It is perpendicular to that edge.
2. The centre-to-edge distance is 6 cm and the perpendicular height is 8 cm.
3. For the acute plane angle θ at the edge midpoint, tan θ = 8/6 = 4/3.

Answer: θ ≈ 53.1°.

Revise first: Angles between lines and planes, Pythagoras in three dimensions.