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

Any three non-collinear points in 3D space define a unique plane. To compute its equation in standard form A x + B y + C z + D = 0, take two edge vectors on the triangle, v₁ = P₂ − P₁ and v₂ = P₃ − P₁, and set the plane’s normal to the cross product n = v₁ × v₂. Then (A,B,C) = n and D = −n · P₁. If you check with any point on the plane, the expression A x + B y + C z + D evaluates to zero (up to rounding).

This tool also accepts a point and a normal vector directly. In that case, the coefficients are (A,B,C) = n and D = −n · P. The result is scale-invariant: multiplying all of A,B,C,D by the same nonzero constant yields the same geometric plane. For a consistent, human-friendly representation, you can turn on the “unit-normal form,” which divides by ‖n‖ so the normal has length 1.

  • Degenerate inputs: identical or collinear points give a zero cross product, so no unique plane exists.
  • Point–normal form: n · (X − P) = 0, useful in CAD and physics.
  • Intercepts (optional exercise): if C ≠ 0, you can solve for z to see the plane’s height function.

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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