# Equation of a Plane Calculator

URL: https://speedycalculator.com/calculators/math/equation-of-a-plane/
Last updated: 2026-10-30

> The equation of a plane calculator finds ax + by + cz = d from a point and a normal vector, or from three points on the plane.

## Inputs

- 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

- a (coefficient of x)
- b (coefficient of y)
- c (coefficient of z)
- d (right-hand side)

## 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)
```

## Explanation

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

FieldWhat to enterDefine the plane byChoose one: A point and a normal vector, Three points P, Q, RP – xA numberP – yA numberP – zA numberNormal aA numberNormal bA numberNormal cA numberQ – xA numberQ – yA numberQ – zA numberR – xA numberR – yA numberR – zA number

### Example values

How the main result changes when **P – x** changes and everything else stays at the example values above:

P – xa (coefficient of x)0.520.752121.2521.5222

## 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.

## FAQ

**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.


## Sources

- [OpenStax – Calculus Volume 3](https://openstax.org/details/books/calculus-volume-3)
- [Wolfram MathWorld – Plane](https://mathworld.wolfram.com/Plane.html)