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Angle Between Planes

The angle between two planes equals the angle between their normal vectors. For normals n₁ and n₂, cos θ = |n₁·n₂|/(|n₁||n₂|). Parallel planes have angle 0°; perpendicular planes 90°.

Concept Fundamentals
cos θ = |n₁·n₂|/(|n₁||n₂|)
Formula
n₁ ∝ n₂ → 0°
Parallel
n₁·n₂ = 0 → 90°
Perpendicular
min(θ, 180°−θ)
Acute

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The D term does not affect the angle—only the normal direction matters. Intersection line direction is n₁×n₂ (cross product). Clamp cos θ to [-1,1] for numerical stability.

Key quantities
cos θ = |n₁·n₂|/(|n₁||n₂|)
Formula
Key relation
n₁ ∝ n₂ → 0°
Parallel
Key relation
n₁·n₂ = 0 → 90°
Perpendicular
Key relation
min(θ, 180°−θ)
Acute
Key relation

Ready to run the numbers?

Why: Dihedral angles between planes appear in crystallography, roof design, molecular geometry, and 3D graphics. The normal-vector approach is the standard method.

How: For plane Ax+By+Cz+D=0, the normal is (A,B,C). Compute cos θ = |n₁·n₂|/(|n₁||n₂|), then θ = arccos(cos θ). Use the acute angle.

The D term does not affect the angle—only the normal direction matters.Intersection line direction is n₁×n₂ (cross product).
Sources:Khan Academy — 3D GeometryWolfram MathWorld — Dihedral Angle

Run the calculator when you are ready.

Calculate AngleEnter normal vectors (A,B,C) for each plane

Sample Examples

Input

Enter normal vectors (A,B,C) for each plane. D does not affect the angle.

Plane 1 Normal (A₁,B₁,C₁)

Plane 2 Normal (A₂,B₂,C₂)

Results

Angle Between Planes

45°

0.79 rad

Step-by-Step

n₁ = (1, 0, 0), |n₁| = 1

n₂ = (1, 1, 0), |n₂| = 1.41

n₁·n₂ = 1·1 + 0·1 + 0·0 = 1

cos θ = 1/(1·1.41) = 0.71

θ = 45° (acute angle)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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cos θ = |n₁·n₂|/(|n₁||n₂|) for plane normals.

— 3D Geometry

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Perpendicular planes: n₁·n₂ = 0.

— Example

Key Takeaways

  • Angle between planes = angle between their normal vectors.
  • costheta=frac∣vecn1cdotvecn2∣∣vecn1∣∣vecn2∣\\cos\\theta = \\frac{|\\vec{n}_1 \\cdot \\vec{n}_2|}{|\\vec{n}_1||\\vec{n}_2|}.
  • Parallel planes: normals proportional, angle 0°.
  • Perpendicular planes: vecn1cdotvecn2=0\\vec{n}_1 \\cdot \\vec{n}_2 = 0, angle 90°.
  • We use the acute angle (0° to 90°).

Did You Know?

Dihedral angle in chemistry uses the same formula.
Roof design relies on angles between roof planes.
Crystallography uses plane angles for crystal structure.
Collision response in games uses plane angles.
Molecular geometry studies dihedral angles.
Architecture uses this for structural analysis.

Understanding

The angle between two planes is the angle between their normal vectors. For Ax+By+Cz+D=0, the normal is (A,B,C). The D term does not affect the angle.

costheta=frac∣vecn1cdotvecn2∣∣vecn1∣∣vecn2∣\\cos\\theta = \\frac{|\\vec{n}_1 \\cdot \\vec{n}_2|}{|\\vec{n}_1||\\vec{n}_2|}
vecn1cdotvecn2=A1A2+B1B2+C1C2\\vec{n}_1 \\cdot \\vec{n}_2 = A_1A_2 + B_1B_2 + C_1C_2

Expert Tips

Use acute angle: min(θ, 180°-θ).
Clamp cos to [-1,1] for numerical stability.
Parallel: |cos θ| ≈ 1.
Perpendicular: cos θ ≈ 0.

FAQ

Q: Why use normals?
A: Normals define plane orientation; their angle = plane angle.
Q: What if normals are zero?
A: Invalid plane; A,B,C not all zero.
Q: Obtuse or acute?
A: Convention: acute angle (0°–90°).
Q: Coincident planes?
A: Angle 0°, normals parallel.
Q: Applications?
A: Roof design, crystallography, robotics, graphics.
Q: Relation to line of intersection?
A: Direction of intersection = n₁×n₂.
Q: Can angle exceed 90°?
A: We report acute angle; max 90°.

How to Use

  1. Enter plane 1 normal: (A₁, B₁, C₁).
  2. Enter plane 2 normal: (A₂, B₂, C₂).
  3. Get angle in degrees and radians.

Disclaimer

Normal vectors (A,B,C) must be non-zero for each plane. D does not affect the angle.

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