Point to Plane Distance Calculator

About this calculator

The perpendicular distance from a point (x₀, y₀, z₀) to a plane defined by ax + by + cz + d = 0 has a closed-form solution: d = |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²). The denominator is the magnitude of the plane’s normal vector (a, b, c), and dividing by it normalizes the numerator into a true geometric distance rather than an arbitrary scaled value.

This formula is a workhorse in 3D computer graphics and game engines, where it drives collision detection and view-frustum clipping — checking whether an object has crossed a wall or camera boundary reduces to exactly this calculation. CAD/CAM software and robotics path planners use it to verify tolerances, such as confirming that a machined surface point stays within a required distance of a reference plane, and structural engineers use it to check clearances between a load point and a wall or floor plane during design review.

Enter the plane’s coefficients (a, b, c, d) and the point’s coordinates, and this calculator returns the shortest, perpendicular distance between them.