Calculators / Calculus 3 / Angle Between Two Planes (3D)

Angle Between Two Planes (3D)

Published on: June 14, 2026

This calculator finds the angle between two planes in 3D. Enter input like Ax+By+Cz=d; A2x+B2y+C2z=d2. Example: 2x-3y+4z=10; x+4y-2z=7.

Step-by-step method

  1. Identify the normal vectors n₁ and n₂ from the two planes.
  2. Compute the dot product |n₁ · n₂|.
  3. Compute the magnitudes |n₁| and |n₂|.
  4. Use cos(θ) = |n₁ · n₂| / (|n₁||n₂|) and solve for θ.

Formula:

Ax + By + Cz = d
For Ax+By+Cz=d, normal n = <A, B, C>
cos(θ) =
|n₁ · n₂|
|n₁||n₂|

Example 1: 2x-3y+4z=10; x+4y-2z=7

Step 1 - Identify the two plane normals n₁ and n₂.

In this problem: For each plane Ax+By+Cz=d, the normal is n=<A,B,C>.

Plane 1: A₁=2, B₁=-3, C₁=4, d₁=10
n₁ = < 2, -3, 4 >
Plane 2: A₂=1, B₂=4, C₂=-2, d₂=7
n₂ = < 1, 4, -2 >

Step 2 - Compute |n₁·n₂|.

In this problem: Compute the dot product and take absolute value.

|n₁ · n₂| = |2·1 + -3·4 + 4·-2| = |-18| = 18

Step 3 - Compute |n₁| and |n₂|.

In this problem: Use |n| = √(A²+B²+C²).

|n₁| = √22 + -32 + 42 = √29
|n₂| = √12 + 42 + -22 = √21

Step 4a - Substitute solved values into cos(θ).

In this problem: Use cos(θ)=|n₁·n₂|/(|n₁||n₂|) with the computed numbers.

cos(θ) =
|n₁ · n₂|
|n₁||n₂|
=
18
√29 √21

Step 4b - Solve for θ.

In this problem: Take arccos to get the angle between the planes.

θ = arccos(
18
√29 √21
) ≈ 43.16° (≈ 0.75 rad)

Final answer: Angle = 180*acos(6*sqrt(609)/203)/pi degrees

Example 2: x+y+z=6; 2x-y+2z=7

Step 1 - Identify the two plane normals n₁ and n₂.

In this problem: For each plane Ax+By+Cz=d, the normal is n=<A,B,C>.

Plane 1: A₁=1, B₁=1, C₁=1, d₁=6
n₁ = < 1, 1, 1 >
Plane 2: A₂=2, B₂=-1, C₂=2, d₂=7
n₂ = < 2, -1, 2 >

Step 2 - Compute |n₁·n₂|.

In this problem: Compute the dot product and take absolute value.

|n₁ · n₂| = |1·2 + 1·-1 + 1·2| = |3| = 3

Step 3 - Compute |n₁| and |n₂|.

In this problem: Use |n| = √(A²+B²+C²).

|n₁| = √12 + 12 + 12 = √3
|n₂| = √22 + -12 + 22 = √9

Step 4a - Substitute solved values into cos(θ).

In this problem: Use cos(θ)=|n₁·n₂|/(|n₁||n₂|) with the computed numbers.

cos(θ) =
|n₁ · n₂|
|n₁||n₂|
=
3
√3 √9

Step 4b - Solve for θ.

In this problem: Take arccos to get the angle between the planes.

θ = arccos(
3
√3 √9
) ≈ 54.74° (≈ 0.96 rad)

Final answer: Angle = 180*acos(sqrt(3)/3)/pi degrees