Planes are the 3D equivalent of lines. Their equations are used in computer graphics (clipping and lighting), surveying, crystallography and physics.
How to use the Equation of a Plane Calculator
- Define the plane by
- P – x
- P – y
- P – z
- Normal a
- Normal b
- Normal c
- Q – x
- Q – y
- Q – z
- R – x
- R – y
- R – z
Results update as you type, and everything is calculated in your browser.
How it works
With three points, two edge vectors are formed and their cross product gives the normal. d is found by substituting a point into the equation.
Inputs explained
| Field | What to enter |
|---|---|
| Define the plane by | Choose one: A point and a normal vector, Three points P, Q, R |
| P – x | A number |
| P – y | A number |
| P – z | A number |
| Normal a | A number |
| Normal b | A number |
| Normal c | A number |
| Q – x | A number |
| Q – y | A number |
| Q – z | A number |
| R – x | A number |
| R – y | A number |
| R – z | A number |
Example values
How the main result changes when P – x changes and everything else stays at the example values above:
| P – x | a (coefficient of x) |
|---|---|
| 0.5 | 2 |
| 0.75 | 2 |
| 1 | 2 |
| 1.25 | 2 |
| 1.5 | 2 |
| 2 | 2 |
Formula
Point P and normal n = ⟨a, b, c⟩: a(x − p_x) + b(y − p_y) + c(z − p_z) = 0
Three points: n = (Q − P) × (R − P)
Worked example
Inputs — Define the plane by: A point and a normal vector; P – x: 1; P – y: 2; P – z: 3; Normal a: 2; Normal b: -1; Normal c: 4.
Results — a (coefficient of x) = 2; b (coefficient of y) = -1; c (coefficient of z) = 4; d (right-hand side) = 12.
Key terms
- Plane
- A flat 2D surface extending infinitely in 3D space.
Frequently asked questions
What does the normal vector tell me?
The direction perpendicular to the plane; its components are a, b and c.
What if the three points are collinear?
The cross product is zero and no unique plane exists.
How do I check a point is on the plane?
Substitute it: ax + by + cz must equal d.
How do I write the plane in intercept form?
x/(d/a) + y/(d/b) + z/(d/c) = 1 when a, b, c, d ≠ 0.