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Point to Plane Distance Calculator

Distance from a point to a plane.

How it is calculated

The plane is entered in general form a·x + b·y + c·z + d = 0 through the coefficients a, b, c, d, and the point as x0, y0, z0. The tool computes distance = |a·x0 + b·y0 + c·z0 + d| / √(a² + b² + c²), rounded to 4 decimals. The absolute value means the result is never negative, so it does not show which side of the plane the point is on. All seven fields must be filled (0 is allowed); a blank one gives no result, and so does the normal (a, b, c) = (0, 0, 0), which does not describe a plane. A distance of 0 means the point lies on the plane.

Worked example

Plane 2x + 3y + 6z − 14 = 0 and point (1, 2, 3). Substitute the point: 2 × 1 + 3 × 2 + 6 × 3 − 14 = 2 + 6 + 18 − 14 = 12. The normal's length is √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7. Divide: 12 / 7 = 1.7143 units. The point is therefore about 1.71 units from the plane, measured along the shortest, perpendicular line.

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Common mistakes

  • Leaving the plane in the form ax + by + cz = e and typing e as d. The tool expects everything on the left with zero on the right, so d = −e (here d = −14 for 2x + 3y + 6z = 14).
  • Forgetting the sign of d. Using +14 instead of −14 changes the numerator to 40 and the distance to about 5.7143.
  • Leaving a coefficient blank when it is zero. For a plane such as x + 2z = 4, type 0 for b rather than clearing the field.
  • Expecting a signed result. The absolute value hides which side the point is on; compute the sign of the numerator yourself if you need that.

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.

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