What is the formula for the distance from a point to a plane?

For a plane ax + by + cz + d = 0 and a point P(x0, y0, z0), the distance is D = |ax0 + by0 + cz0 + d| / sqrt(a² + b² + c²). The absolute value ensures the result is always non-negative.

What does it mean if the distance is zero?

A distance of zero means the point lies exactly on the plane. In other words, the point satisfies the plane equation ax0 + by0 + cz0 + d = 0.

What if a, b, and c are all zero?

If a = b = c = 0 the equation does not define a valid plane (the normal vector has zero magnitude). The calculator will display an error in this case.

What is the foot of the perpendicular?

The foot of the perpendicular is the unique point on the plane that is closest to the given point. It lies along the line through the point in the direction of the plane's normal vector (a, b, c).

Can I enter negative coefficients?

Yes. All four plane coefficients (a, b, c, d) and all three point coordinates can be any real number, including negatives and decimals.

How is this different from the distance between two points?

Point-to-point distance measures along a straight line between two specific locations. Point-to-plane distance measures along the perpendicular from the point to the nearest spot on an infinite flat surface.

Point To Plane Distance Calculator

Name: Point To Plane Distance Calculator
Availability: InStock
Author: Nham Vu

Enter the plane coefficients (a, b, c, d) and a 3D point (x, y, z) to instantly calculate the shortest perpendicular distance.

Plane & Point

Plane equation: ax + by + cz + d = 0

Plane Coefficients

Point Coordinates

Quick Examples

Enter coefficients and a point, then click Calculate Distance.

Copied!

Summary

Enter the plane coefficients (a, b, c, d) and a 3D point (x, y, z) to instantly calculate the shortest perpendicular distance.

How it works

Enter the plane coefficients a, b, c, and d for the equation ax + by + cz + d = 0.
Enter the coordinates x, y, z of the point in 3D space.
The calculator evaluates the numerator: |ax0 + by0 + cz0 + d|.
It computes the denominator: the magnitude of the normal vector sqrt(a² + b² + c²).
The distance is their ratio, displayed with a full step-by-step breakdown.
The foot of the perpendicular (closest point on the plane) is also shown.

Use cases

Solving analytic geometry homework problems involving 3D planes.
Checking shortest distances in CAD and 3D modeling calculations.
Physics simulations needing point-to-surface proximity.
Computer graphics and ray-tracing distance queries.
Robotics path planning where obstacles are modeled as planes.
Verifying hand-calculated results for distance-to-plane problems.
Engineering applications involving structural planes and load points.
Teaching and learning 3D coordinate geometry concepts.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu