When is distance between two planes nonzero?

Only when the planes are parallel and distinct. Intersecting or coincident planes have minimum distance 0.

How do I know if two planes are parallel?

Their normal vectors must be collinear, meaning one normal is a scalar multiple of the other.

Why does the calculator scale one equation?

Distance between parallel planes requires both equations to use the same normal vector before comparing constant terms.

Does the angle between planes affect distance?

If the angle is not 0, the planes intersect, so the shortest distance is 0.

Distance Between Planes Calculator

Calculate the distance between two planes in 3D by checking whether they are parallel, coincident, or intersecting, then measuring parallel-plane separation.

Examples

First plane: equation coefficients

x coefficient

y coefficient

z coefficient

constant term

Second plane: equation coefficients

x coefficient

y coefficient

z coefficient

constant term

The calculator first checks whether the normal vectors are parallel. Intersecting planes have distance 0; parallel planes compare the constant terms after scaling.

Distance between parallel planes

English results for distance between planes focus on the parallel case. If two planes intersect, their minimum distance is 0; if they are coincident, they are the same plane.

d = \frac{∣ D _{1} - D _{2}^{'} ∣}{A ^{2} + B ^{2} + C ^{2}}

D2' is the second plane's constant term after scaling its normal to match the first plane.

Classification before distance

Non-collinear normals mean the planes intersect, so the shortest distance is 0.
Collinear normals with matching scaled constants mean the planes coincide.
Collinear normals with different scaled constants mean the planes are parallel and separated.
The distance uses the same coordinate units as the plane equations.

Examples

Plane 1	Plane 2	Result
x + y + z -	x + y + z -	parallel; distance about 1.732
x + y + z -	2x + 2y + 2z -	coincident; distance 0
x +	x -	intersecting; distance 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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