Point To Plane Distance Calculator

Work out point to plane distance instantly with clear inputs, formula shown and shareable results.

About Point To Plane Distance Calculator

For the plane ax + by + cz + d = 0 the perpendicular distance from a point is |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²). Keeping the sign tells you which side of the plane the point lies on.

Formula

Point-plane distance

D = |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²)

Worked example

With the default values below, Point To Plane Distance Calculator returns Perpendicular distance of 1.2857. Change any input above and every figure here is recalculated in your browser.

Inputs used

Plane coefficients a, b, c, d: 2, -3, 6, -5
Point (x, y, z): 1, 2, 3

Results

Perpendicular distance: 1.2857
Signed distance: 1.2857
|n| = √(a² + b² + c²): 7.0000
Which side: On the positive side of the plane

Step by step

1Values used
2Point-plane distance
3Perpendicular distance= 1.2857
4Signed distance= 1.2857
5|n| = √(a² + b² + c²)= 7.0000
6Which side= On the positive side of the plane

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Plane 2x − 3y + 6z − 5 = 0 and point (1,2,3)?

The numerator is |2 − 6 + 18 − 5| = 9 and |n| = 7, so the distance is 9/7 ≈ 1.285714.

What sign convention applies?

The sign follows the direction of the normal vector (a, b, c); reversing all four coefficients flips the sign but not the distance.