Point-to-Plane Distance Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The distance from a point to a plane in space is calculated with d = |ax₀+by₀+cz₀+d| ÷ √(a²+b²+c²), where ax+by+cz+d=0 is the plane's equation and (x₀,y₀,z₀) the point's coordinates. For the plane x+y+z−6=0 and the point (1,2,3), which lies exactly on that plane, the distance is 0.
Explanation
This formula directly generalizes to three dimensions our point-to-line distance calculator in two dimensions: the numerator evaluates the plane's equation at the given point (a zero result meaning the point lies exactly on the plane, as this calculator's first test case confirms), while the denominator normalizes that result by the norm of the plane's normal vector (a,b,c) — the vector perpendicular to the plane that determines its orientation in space. This same mathematical construction — evaluate the equation of a geometric figure at a point, then divide by the norm of the direction coefficients — appears systematically whenever you seek the shortest distance between a point and a line or a plane, whatever the dimension of the space considered. This distance always represents the shortest possible distance between the point and any point of the plane, measured perpendicular to it — it is the shortest segment connecting the point to the plane, never a longer oblique path.
Example: plane 2x+3y+6z−12=0, point at the origin
Inputs
Plane: 2x + 3y + 6z − 12 = 0. Point: (0, 0, 0).
Calculation
d = |2×0 + 3×0 + 6×0 − 12| ÷ √(2² + 3² + 6²) = 12 ÷ √49 = 12 ÷ 7 ≈ 1.7143.
Result
The distance between the origin and this plane is about 1.7143 units.
Frequently asked questions
What does a distance of 0 mean?
A distance of exactly zero means the point lies precisely on the plane: its coordinates, once substituted into the plane's equation, satisfy that equation exactly (the numerator of the formula then vanishes completely), as in this calculator's first test case, where the point (1,2,3) satisfies x+y+z=6 exactly.
What do the coefficients a, b, and c of the plane's equation represent?
These three coefficients form the plane's normal vector — the vector perpendicular to its surface, which determines its orientation in space. Two parallel planes share the same normal vector (up to a scalar factor), while two planes with different normal vectors intersect along a line.
How do you get the plane's equation if you don't already know it?
A plane's equation can be determined from three non-collinear points on it, or from a point on the plane combined with its normal vector, using the cross product of two vectors within the plane to get exactly that normal vector — see our 3D cross product calculator for that intermediate step.