How do you find the angle between two planes?

Use the dot product of the two normal vectors, then take the arccosine of the normalized absolute value.

Why is the answer between 0 and 90 degrees?

Using the absolute dot product removes the arbitrary direction of the normals and returns the smaller angle.

Does D affect the angle between planes?

No. D shifts a plane but does not change its normal direction.

Is this the same as a dihedral angle?

For analytic geometry calculators, yes: it is the smaller angle between the two planes.

Angle Between Planes Calculator

Calculate the smaller angle between two planes from their normal vectors or standard plane equations, with degree/radian output and relationship labels.

Plane 1: general equation coefficients

A₁

B₁

C₁

D₁

D changes the plane position, not the angle.

Plane 2: general equation coefficients

A₂

B₂

C₂

D₂

D changes the plane position, not the angle.

Method: plane normals, dot product, and the smaller dihedral angle from 0° to 90°.

Angle between two planes

Search results for angle between planes use the angle between normal vectors to calculate the smaller dihedral angle. The constant terms D shift planes but do not change the angle.

cos θ = \frac{∣ n _{1} \cdot n _{2} ∣}{∥ n _{1} ∥ ∥ n _{2} ∥}

theta is the smaller angle between the planes; n1 and n2 are their normal vectors.

How to read the result

0° means the planes are parallel or coincident.
90° means the planes are perpendicular.
A value between 0° and 90° means the planes intersect at an acute angle.
The calculator reports both degrees and radians.

Because a plane normal can point in either direction, the smaller analytical angle is usually reported.

Examples

Plane 1	Plane 2	Angle
x + y + z -	x -	90°
	x +	45°
x + y +	2x + 2y + 2z -	0°

Frequently Asked Questions

Sources and References

Calculations are based on the listed reference sources. Links open in a new tab.

Updated: June 2, 2026

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