LifeTally
Point-to-plane distance and closest point in 3D
How far is your point from the plane, and where is the perpendicular foot?
Enter a plane Ax + By + Cz = D and a point P to find its distance and the three coordinates of the closest point Q.
Your plane and point
D is the right-hand side of Ax + By + Cz = D. Use the same perpendicular Cartesian axes, equal scale and one coordinate-unit basis. All seven accepted values are required, even when a coefficient is zero.
How the estimate works
The normal vector is N = (A, B, C). With S = A² + B² + C² and r = A·Px + B·Py + C·Pz − D, the closest point is Q = P − (r/S)N and the distance is √(r²/S). Q lies on the plane and P − Q is perpendicular to it. A point already on the plane has distance zero and Q = P.
Ax + By + Cz = D; N = (A, B, C)
S = A² + B² + C²; r = A·Px + B·Py + C·Pz − D
Q = P − (r ÷ S)N; d = √(r² ÷ S)Only one point and one complete infinite Euclidean plane given by coefficients. This is not distance to a finite face, triangle, polygon, segment, curved surface, solid or mesh. No signed distance, side classification, plane fitting, graph, geographic coordinates, unit conversion, measurement validation, uncertainty, collision test or physical-fit or safety guarantee. Check that all four coefficients describe the intended equation in the chosen coordinate units.
OpenStax: plane equations and perpendicular distance ↗Check the filled example
Our chosen example: plane 2x − y + 2z = 6, point P = (4, 1, 4). S = 9, r = 9 and r/S = 1, so Q = (2, 2, 2) and distance = 3. Substitution gives 2·2 − 2 + 2·2 = 6. These are mathematical example values, not measured or recommended dimensions.
Accepted input values
- Plane: A, coefficient of x
- 2
- Plane: B, coefficient of y
- -1
- Plane: C, coefficient of z
- 2
- Plane: D, right-hand side
- 6
- Point P: x
- 4
- Point P: y
- 1
- Point P: z
- 4
Plan the next step
Use the standard calculator to check arithmetic steps. Check the plane equation, the coordinate axes and the unit basis before interpreting the result.
Standard calculator →