The shortest distance from a point to a plane is measured along the plane’s normal. It is used in 3D graphics (clipping and collision), robotics, surveying and crystallography.
How to use the Distance from Point to Plane Calculator
- Point a (x)
- Point b (y)
- Point c (z)
- How is your plane given?
- A
- B
- C
- D
- Point on plane x
- Point on plane y
- Point on plane z
Results update as you type, and everything is calculated in your browser.
How it works
The point is substituted into the plane equation and the result divided by the length of the normal vector. With a normal and a point, D is computed first.
Inputs explained
| Field | What to enter |
|---|---|
| Point a (x) | A number |
| Point b (y) | A number |
| Point c (z) | A number |
| How is your plane given? | Choose one: Standard form Ax + By + Cz + D = 0, Normal vector and a point on the plane |
| A | A number |
| B | A number |
| C | A number |
| D | A number |
| Point on plane x | A number |
| Point on plane y | A number |
| Point on plane z | A number |
Example values
How the main result changes when Point a changes and everything else stays at the example values above:
| Point a (x) | Distance from point to plane |
|---|---|
| 0.5 | 0.3333 |
| 0.75 | 0.5 |
| 1 | 0.6667 |
| 1.25 | 0.8333 |
| 1.5 | 1 |
| 2 | 1.3333 |
Formula
L = |A·a + B·b + C·c + D| ÷ √(A² + B² + C²)
Worked example
Inputs — Point a (x): 1; Point b (y): 2; Point c (z): 3; How is your plane given?: Standard form Ax + By + Cz + D = 0; A: 2; B: -1; C: 2; D: -4.
Results — Distance from point to plane = 0.66666667; Signed distance = 0.66666667; Length of the normal |n| = 3.
Key terms
- Normal vector
- A vector perpendicular to a plane.
Frequently asked questions
Why divide by √(A² + B² + C²)?
To turn the plane equation value into a true distance along a unit normal.
What does the sign of the signed distance mean?
Which side of the plane the point is on.
What if the distance is zero?
The point lies on the plane.
How do I get D from a normal and a point?
D = −(A·x₀ + B·y₀ + C·z₀).