Calculates the shortest (perpendicular) distance from a point in 3D space to a plane.
How it works
The point’s coordinates are substituted into the plane’s equation, and the result is divided by the magnitude of the plane’s normal vector.
What this does not include
This does not include the distance between two points — for that simpler 3D calculation, use this site’s 3D distance calculator instead.
How to use this calculator
- Enter the plane’s equation coefficients (A, B, C, D) and the point’s coordinates.
A worked example
Plane x + 2y − 2z = 3, point (3, 1, 2): distance = |1(3)+2(1)−2(2)−3| / √(1²+2²+2²) = 0.6667.
Plane 2x − y + 2z + 4 = 0, point at the origin (0,0,0): distance = 1.3333.
What the variables mean
| Variable | Meaning |
|---|---|
| a, b, c, d | Plane coefficients, from ax + by + cz + d = 0 |
| (x0, y0, z0) | The point being measured from |
Edge cases worth knowing
The three plane coefficients (a, b, c) can’t all be zero — without at least one, the equation no longer describes a valid plane, so the calculator declines to show a result.
A distance of zero means the point lies exactly on the plane — a valid and meaningful result, not a sign of a broken calculation.
Frequently asked questions
What does it mean for A, B, and C to all be zero?
The equation no longer describes a valid plane — those three coefficients form the plane’s normal vector, and a zero vector has no direction.
What if the point lies exactly on the plane?
The distance comes out to exactly zero, since the point satisfies the plane’s equation with no leftover value.
What’s a real-world use for this calculation?
Collision detection in 3D graphics and games, and finding clearances in engineering designs, both rely on point-to-plane distance calculations.
