Angle Between Two Planes Calculator

This calculator finds the angle between two planes. Enter the coefficients a, b and c of each plane written as ax + by + cz = d (the d values do not matter); you get the acute angle between the planes in degrees and its cosine.

Formula

cos θ = |n₁ · n₂| ÷ (|n₁| |n₂|), where n₁ = (a₁, b₁, c₁) and n₂ = (a₂, b₂, c₂) are the normal vectors of the planes.

n₁ · n₂ = a₁a₂ + b₁b₂ + c₁c₂ (the dot product)
|n| = √(a² + b² + c²)
The absolute value gives the acute angle, between 0° and 90°.

Worked Example

Plane 1: x + 2y + 2z = 5. Plane 2: 2x − y + 2z = 1.

n₁ · n₂ = 1 × 2 + 2 × (−1) + 2 × 2 = 4
|n₁| = √(1 + 4 + 4) = 3 and |n₂| = √(4 + 1 + 4) = 3
cos θ = 4 ÷ 9 = 0.4444
θ = arccos 0.4444 = 63.6122°

How to read the result

0° means the planes are parallel (or the same plane); 90° means they are perpendicular. The obtuse angle between the planes, if you need it, is 180° minus the result.

Limits and common mistakes

Read a, b and c from the equation in the form ax + by + cz = d. Move every term to one side first.
A plane cannot have a = b = c = 0; that gives Cannot compute.
The angle between a line and a plane uses sin, not cos.

Frequently Asked Questions

When are two planes perpendicular?

When the dot product of their normal vectors is 0.

When are they parallel?

When the normal vectors are multiples of each other; the angle is then 0°.

Does d change the angle?

No. d only shifts the plane; the direction comes from a, b and c.

Can I get the obtuse angle?

Yes: 180° minus the acute angle shown.

Formula used

in a1 | Plane 1, a | - | 1 | -
in b1 | Plane 1, b | - | 2 | -
in c1 | Plane 1, c | - | 2 | -
in a2 | Plane 2, a | - | 2 | -
in b2 | Plane 2, b | - | -1 | -
in c2 | Plane 2, c | - | 2 | -
out ang | Angle between the planes | ° | acosd(abs(a1 * a2 + b1 * b2 + c1 * c2) / (sqrt(a1^2 + b1^2 + c1^2) * sqrt(a2^2 + b2^2 + c2^2))) | 4
out cs | cos θ | - | abs(a1 * a2 + b1 * b2 + c1 * c2) / (sqrt(a1^2 + b1^2 + c1^2) * sqrt(a2^2 + b2^2 + c2^2)) | 4

Each line is one input (in, sel, list) or one result (out: label, unit, formula, decimals). How the calculators work

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Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.