Distance from a Point to a Plane
The distance between a point P and a plane is the length of the perpendicular segment connecting point P to the plane.
In other words, the distance from a point P to a plane is the length of the perpendicular line segment that extends from point P to the plane.
The distance is always a non-negative value. If the point lies on the plane, the distance is zero.
How to Calculate the Distance Between a Point and a Plane
The shortest distance between a point P and a plane α in space is the length of the perpendicular segment from P to the plane α.
$$ d(P, \alpha ) = \frac{|ax_0+by_0+cz_0+d|}{\sqrt{a^2+b^2+c^2}} $$
Example
Consider point P with coordinates (x;y;z) = (1;3;2) in three-dimensional space:
$$ P = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} $$
The Cartesian equation of the plane is:
$$ 4x-2y+z-5 = 0 $$
The distance between the point and the plane is calculated using the formula above.
$$ d(P, \alpha) = \frac{|ax_0+by_0+cz_0+d|}{\sqrt{a^2+b^2+c^2}} $$
The parameters of the plane equation are a=4, b=-2, c=1, and d=-5:
$$ d(P, \alpha) = \frac{|4 \cdot x_0 - 2 \cdot y_0 + 1 \cdot z_0 - 5|}{\sqrt{4^2 + (-2)^2 + 1^2}} $$
$$ d(P, \alpha) = \frac{|4 \cdot x_0 - 2 \cdot y_0 + 1 \cdot z_0 - 5|}{\sqrt{21}} $$
Substituting the coordinates of the point x0=1, y0=3, z0=2:
$$ d(P, \alpha) = \frac{|4 \cdot 1 - 2 \cdot 3 + 1 \cdot 2 - 5|}{\sqrt{21}} $$
$$ d(P, \alpha) = \frac{|-5|}{\sqrt{21}} $$
$$ d(P, \alpha) = \frac{5}{\sqrt{21}} = 1.09 $$
Therefore, the distance between point P and the plane is 1.09.
In the following graphic, this is represented by the red line between point P and the plane.
It is the length of the perpendicular segment from point P to the plane.
And so on.
