What this calculator does
The distance from a point to a plane is the same idea as point to line, one dimension up. Substitute the point into the plane equation, take the absolute value, and divide by the length of the normal vector.
Dividing by the normal length is what makes the answer a real distance. Without it the result would depend on how the equation happened to be scaled, since 2x − y + 2z − 6 = 0 and 4x − 2y + 4z − 12 = 0 describe the same plane.
The formula
The point coordinates are substituted into the plane equation. The absolute value of the result is divided by the square root of the sum of the squared coefficients.
| Term | Meaning |
|---|---|
| Plane equation | ax + by + cz + d = 0, the general form in three dimensions. |
| Normal vector | The vector (a, b, c), which points perpendicular to the plane. |
| Signed value | The substitution before taking the absolute value, whose sign gives the side. |
The inputs explained
| Field | What to enter |
|---|---|
| Plane: a | Plane coefficient of x. |
| Plane: b | Plane coefficient of y. |
| Plane: c | Plane coefficient of z. The three cannot all be zero. |
| Plane: d | The plane constant. |
| Point: x₀ | Point x coordinate. |
| Point: y₀ | Point y coordinate. |
| Point: z₀ | Point z coordinate. |
When to use it
Measuring clearance in 3D
The perpendicular distance is the shortest gap to a flat surface.
Collision detection
Point to plane distance is a core primitive in graphics and physics engines.
Testing whether a point lies on a plane
A distance of zero settles it exactly.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does moving the point change the distance?
Three points measured against the same plane.
| Point x (y = 0, z = 0) | Distance | Signed value (before |·| and ÷ norm) |
|---|---|---|
| (0, 0, 0) | 2.000 | -6.000 |
| (3, 0, 0) | 0 | 0 |
| (9, 0, 0) | 4.000 | 12.000 |
Questions
Why divide by the normal length?
To normalise the equation. The same plane can be written with any scalar multiple of its coefficients, and without that division the answer would change depending on which multiple was used.
What does the sign of the substitution mean?
Which side of the plane the point is on. Two points giving opposite signs sit on opposite sides, which is a cheap way to test that without computing distances.
How do I get the plane equation from three points?
Take two vectors lying in the plane and cross them to get the normal. Those components give a, b and c, and substituting any one of the points gives d.
Does the same formula work in higher dimensions?
Yes, unchanged in structure. A hyperplane in any number of dimensions has the same equation form, and the distance formula generalises directly with more terms.
For the two-dimensional case, see the point to line distance calculator. For distance between points, see the 3D distance calculator.