Distance from Point to Plane Calculator
Enter the plane equation ax + by + cz + d = 0 and the point (x₀, y₀, z₀) to compute the shortest distance and the foot of the perpendicular.
Perpendicular (shortest) distance from the point to the plane
- 1
Normal vector magnitude √(a²+b²+c²)
√(1² + 2² + 2²) = 3 - 2
Plug the point in: ax₀+by₀+cz₀+d
1×4 + 2×1 + 2×-1 + (-6) = -2 - 3
Absolute value (numerator)
|-2| = 2The sign indicates which side of the plane the point is on; distance is always non-negative. - 4
Distance = numerator ÷ |n|
2 ÷ 3 = 0.6667
How does this calculator work?
The perpendicular distance from point (x₀,y₀,z₀) to plane ax+by+cz+d=0 is d = |ax₀+by₀+cz₀+d| / √(a²+b²+c²). Enter the four plane coefficients and three point coordinates; the tool computes the distance, the normal magnitude, and the foot of the perpendicular.
Formula
How this is calculated
A plane in 3D space is defined by the equation ax + by + cz + d = 0, where the vector n = (a, b, c) is called the normal — it is perpendicular to every line that lies in the plane. The shortest distance from an external point P = (x₀, y₀, z₀) to the plane is measured along this normal direction.
The formula d = |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²) follows from projecting the displacement vector from any point Q on the plane to P onto the unit normal. The numerator |ax₀ + by₀ + cz₀ + d| is the absolute value of the signed distance scaled by the normal magnitude; dividing by √(a² + b² + c²) normalises it to actual length units.
The foot of the perpendicular — the closest point on the plane to P — is found by walking back along the normal: foot = P − t·n, where t = (ax₀ + by₀ + cz₀ + d) / (a² + b² + c²). The result is exact for any plane that has a non-zero normal (a, b, c ≠ all zero). If all coefficients a, b, c are zero the equation no longer defines a plane and the calculator returns no result.
Frequently asked questions
A distance of 0 means the point lies on the plane — ax₀ + by₀ + cz₀ + d = 0 is satisfied exactly.
Bring all terms to one side: ax + by + cz + d = 0. For example, the plane 2x − y + 3z = 5 becomes 2x − y + 3z − 5 = 0, so a=2, b=−1, c=3, d=−5.
It is the unique point on the plane that is closest to the given point. The segment from the point to its foot is perpendicular to the plane.
Also known as
TG we-Calculate Editorial Team. (2026). Distance from Point to Plane Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/distance-from-point-to-plane-calculator
TG we-Calculate Editorial Team. "Distance from Point to Plane Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/distance-from-point-to-plane-calculator.
TG we-Calculate Editorial Team, "Distance from Point to Plane Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/distance-from-point-to-plane-calculator
@misc{wecalculate_distance_from_point_to_plane_calculator, title = {Distance from Point to Plane Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/distance-from-point-to-plane-calculator}}, year = {2026}, note = {TG we-Calculate} }
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