Distance From Point To Plane Calculator

Distance from Point to Plane Calculator

Calculate the perpendicular distance from a point to a plane in 3D space.

Last updated: April 2026 | By Patchworkr Team

3D Geometry

Result will appear here...

What is Distance from Point to Plane?

The distance from a point to a plane is the length of the perpendicular line segment from the point to the plane.

  • Formula: d = |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²)
  • Plane equation: ax + by + cz + d = 0
  • Point: (x₀, y₀, z₀)

This is the shortest distance between the point and any point on the plane.

How to Calculate

  1. Identify the point coordinates (x₀, y₀, z₀)
  2. Identify the plane coefficients a, b, c, d from equation ax + by + cz + d = 0
  3. Substitute into the distance formula
  4. Calculate the perpendicular distance

Example

Point:
(1, 2, 3)
Plane:
2x + 3y + 4z + 5 = 0
Result:
≈ 2.41 units

FAQ

What if the distance is zero?

The point lies on the plane.

Is the distance always positive?

Yes, we use absolute value in the formula.

Can this be used in 2D?

Yes, treat z = 0 and use the 2D plane equation ax + by + d = 0.

What does the plane equation mean?

It defines all points (x, y, z) that satisfy ax + by + cz + d = 0.

How is this used in practice?

CAD, robotics, graphics, and physics simulations use this for collision detection.

What if a, b, c are all zero?

This is invalid - you don't have a plane. At least one coefficient must be non-zero.

Does order of points matter?

No, distance is the same regardless of which point or plane orientation.

Can the distance be negative?

No, the absolute value ensures it's always positive.

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