Plane from 3 Points — Plane Equation

Enter three non-collinear points (or a point and a normal) to get Ax + By + Cz + D = 0. Private by design—everything runs locally in your browser.

Inputs & Actions

P₁, P₂, P₃ must be non-collinear (not all on one line).

A x + B y + C z + D = —

Method: n = (P₂−P₁) × (P₃−P₁), then (A,B,C)=n and D = −n·P₁.

Preview

The blue patch shows the plane; black dots are your points; the teal arrow is the plane’s normal.

How to Get the Plane Through Three Points

This calculator finds the equation of a plane in 3D space from three points or from a point and a normal vector. If you have coordinates from geometry homework, CAD modeling, or 3D data, it turns them into a clean plane equation you can use for further calculations, visualization, or validation.

Plain-language idea

A plane is a flat surface that extends forever. In 3D space, any three points that are not on the same line define exactly one plane—like a tabletop resting on three legs. The standard equation is A x + B y + C z + D = 0. The values A, B, and C describe the plane’s orientation, and D shifts the plane to pass through your points.

How the calculation works

The tool builds two vectors from your points: v₁ = P₂ − P₁ and v₂ = P₃ − P₁. The cross product n = v₁ × v₂ gives a normal vector perpendicular to the plane. From there, (A,B,C) = n and D = −n · P₁. If you plug any point on the plane into the equation, the result is zero (allowing for rounding). You can also enter a point and a normal directly, which is common in physics and 3D graphics.

How to use this calculator

  1. Choose your input type: three points or point + normal vector.
  2. Enter the coordinates (x, y, z) for each point or for the point and normal.
  3. Click Calculate to get the plane equation and alternate forms.
  4. Use the preview to see the plane patch, points, and the normal direction.

Common uses and examples

Plane equations are used for slicing 3D models, finding intersections, computing distances, and checking alignment in engineering or architecture. In computer graphics, they help with lighting and collision detection. In surveying or geology, a plane can represent a tilted surface or a best-fit layer. For a simple example, the points (0,0,0), (1,0,0), and (0,1,0) define the flat xy-plane, which gives the equation z = 0.

Quick notes

  • Degenerate inputs: identical or collinear points produce no unique plane.
  • Scale-invariant: multiplying A,B,C,D by the same nonzero value keeps the same plane.
  • Unit-normal form: divides by ‖n‖ so the normal vector has length 1.

5 Fun Facts about Planes from Points

Three points make a floor

Any three non-collinear points act like the legs of a wobbly stool—the flat “seat” they define is exactly your plane.

Shape builder

Normal = “thumbs up”

Take two edge arrows from the triangle of points and do a cross product; the result sticks out like a thumbs-up showing the plane’s direction.

Vector trick

Scaling doesn’t change it

Multiply all coefficients A,B,C,D by 10 or 0.5—the plane stays the same. The equation can look different but the sheet of space is identical.

Same plane

Classroom to games

From math class to 3D games, planes let characters stand on floors, walls block movement, and lights bounce—one formula, many worlds.

Everywhere use

Check “flatness” fast

If a fourth point plugged into A x + B y + C z + D gives zero, it lives on the same plane. Tiny numbers mean “almost flat.”

Quick test

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