Angle between a line and a plane
A standard method from Trigonometry in 3-D: the steps and worked examples.
Steps
- From a point P on the line, drop a perpendicular to the plane, meeting it at N (N is the projection of P).
- Join N to the point A where the line meets the plane.
- The angle is ∠PAN, in the right-angled triangle PAN.
Worked examples
Example 1: Angle between a line and a plane: a cuboid ABCDEFGH has base ABCD with AB = 8 cm and BC = 6 cm; E, F, G, H are above A, B, C, D, and the height is 5 cm. Find the angle between the diagonal AG and the base.
projection of G on the base is C, so the angle is ∠GAC
AC = √(8² + 6²) = 10
tan ∠GAC = GC / AC = 5/10
∠GAC ≈ 26.6°Example 2: Angle between two planes (theorem of three perpendiculars): in the same cuboid, find the angle between the plane ABGH and the base.
line of intersection: AB
in the base, BC ⊥ AB
GC ⊥ base and BC ⊥ AB, so GB ⊥ AB (three perpendiculars)
the angle is ∠GBC
tan ∠GBC = 5/6
∠GBC ≈ 39.8°For the whole topic, read Trigonometry in 3-D: key points, common mistakes and how the DSE tests it.