Plane from 3 Points
Find the plane equation Ax + By + Cz + D = 0 from three 3D points.
Overview
The Plane from 3 Points tool finds the equation of the plane in 3D passing through three given points, in the standard form A * x + B * y + C * z + D = 0. It also returns the normal vector and a unit normal, useful when you need lighting calculations or distance-to-plane queries.
It is built for graphics programmers computing surface normals, CAD students finding cross-section planes and physicists working through statics problems where forces act on an arbitrary plane.
How it works
Given three points P1 = (x1, y1, z1), P2, P3, form two edge vectors u = P2 - P1 and v = P3 - P1. The cross product n = u x v is the normal vector and provides the coefficients (A, B, C). Then D = -(A * x1 + B * y1 + C * z1).
If the three points are collinear, u and v are parallel and the cross product is zero — no unique plane exists. The tool flags this rather than dividing by zero. Otherwise the unit normal is n / |n|.
Examples
P1 = (0,0,0), P2 = (1,0,0), P3 = (0,1,0)
→ plane z = 0
P1 = (1,1,1), P2 = (2,3,4), P3 = (5,0,2)
→ ≈ -7x + 1y + 11z - 5 = 0
P1 = (0,0,1), P2 = (1,0,1), P3 = (0,1,1)
→ plane z = 1
FAQ
Is the plane equation unique?
Up to scaling. Multiplying A, B, C, D by any non-zero constant gives the same plane. Normalising by |n| makes the coefficients canonical.
What if the three points are collinear?
Infinitely many planes pass through a line, so no unique plane exists. The tool reports the degenerate case.
Why does the normal direction matter?
In rendering and physics, the normal direction defines "outward" versus "inward." Reverse the cross-product order to flip the sign.
Can I find the distance from another point to this plane?
Yes — distance = |A*x + B*y + C*z + D| / sqrt(A² + B² + C²). The unit normal makes the formula clean.
Is the equation always in standard form?
The calculator reports both the algebraic form Ax + By + Cz + D = 0 and the normal-point form n . (P - P1) = 0 for convenience.