Angle Between Two Planes: A Complete Student Guide

The angle between two planes is defined as the angle between their normal vectors — because tilting a plane tilts its normal by exactly the same amount. If the normals are n1 and n2, then cos θ = |n1 · n2| / (|n1| |n2|).

Two intersecting planes actually form two angles, θ and 180° − θ, like a pair of scissors. By convention we take the acute one, 0° ≤ θ ≤ 90°, which is what the modulus in the formula guarantees.

Key Concepts

• Dihedral angle: the angle between two planes, measured between their normals.
• The normals (a1, b1, c1) and (a2, b2, c2) are read directly from the plane equations a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0.
• Parallel planes: n1 × n2 = 0, meaning the normals are proportional, and θ = 0°.
• Perpendicular planes: n1 · n2 = 0, giving θ = 90°.
• Coincident planes: the normals are proportional AND the constants are in the same ratio — then they are the same plane, not two.
• The constants d1, d2 do not affect the angle at all; only the normals matter.

Formulas

• cos θ = |a1a2 + b1b2 + c1c2| / (√(a1² + b1² + c1²) √(a2² + b2² + c2²)).
• Parallel: a1/a2 = b1/b2 = c1/c2. Perpendicular: a1a2 + b1b2 + c1c2 = 0.
• The acute angle is the reported one; the obtuse supplement 180° − θ is the other dihedral angle.

Worked Example

Find the angle between the planes x + 2y + 2z = 5 and 2x + y − 2z = 3.

1. Normals: n1 = (1, 2, 2), n2 = (2, 1, −2).
2. Dot product: 1×2 + 2×1 + 2×(−2) = 2 + 2 − 4 = 0.
3. |n1| = √(1 + 4 + 4) = 3; |n2| = √(4 + 1 + 4) = 3.
4. cos θ = 0 / 9 = 0, so θ = 90°. The planes are perpendicular.
5. Geometric sense: the line where the planes intersect is perpendicular to both normals, and the normals are themselves at right angles. Correct.

Common Misconceptions

• You need the planes to intersect first. The formula uses only normals, so it works for parallel planes too, giving θ = 0°.
• The angle is unique. Intersecting planes make two dihedral angles, θ and 180° − θ; we report the acute one.
• The constants d1, d2 matter. They shift the planes but never tilt them, so the angle ignores them completely.

Real-World Applications

• Roof design: angles between roof planes and walls.
• Crystallography: angles between crystal faces.
• Geology: measuring dip angles between rock strata.
• CAD: checking perpendicularity of machined surfaces.

Read the normals off the two plane equations and compute cos θ = |n1 · n2| / (|n1| |n2|) — the acute angle is the answer. Parallel means proportional normals; perpendicular means a zero dot product.