Free · Coordinate geometry calculators

Perpendicular Distance from a Point to a Plane

Find shortest distance to a plane given its equation or a normal vector and a point on the plane.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Point (1, 2, 3) is 5 units from plane z − 8 = 0. The closest distance is measured perpendicular to the plane.

Distance = |Ax + By + Cz + D| ÷ √(A² + B² + C²)

Worked example

Enter these known values and leave the other values blank.

Test point x
1
Test point y
2
Test point z
3
Equation A: normal x
0
Equation B: normal y
0
Equation C: normal z
1
Equation D: constant
-8
Distance using plane equation
5

Assumptions and limitations

  • Coordinates use one consistent Cartesian scale. The normal vector (A, B, C) must be nonzero; it need not have unit length.
  • The equation mode uses Ax + By + Cz + D = 0. If the right side of your equation is k, enter D = −k after collecting terms.
  • The normal-and-point mode builds D = −(Ax₀ + By₀ + Cz₀) from the supplied point on the plane.
  • Outputs are unsigned, read-only perpendicular distances. They do not identify which side of the plane contains the point or calculate the closest point’s coordinates.

Common questions

Must I normalize the normal vector first?

No. Dividing by its magnitude makes the result invariant when all plane coefficients are multiplied by the same nonzero scale.

What if the point lies on the plane?

The numerator is zero, so the distance is zero as long as the plane’s normal is nonzero.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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