$Distance from a Point to a Line$

Distance from a Point to a Line

Find the distance from a point to a line in 2D and 3D and from a point to a plane, with the foot of the perpendicular and a step-by-step solution.

What is known:

Point and line Ax + By + C = 0

Point and line through two points

Point and plane Ax + By + Cz + D = 0

Space: in the plane (x, y) in space (x, y, z)

Coordinates are whole numbers, decimals or fractions: 3, −1.5, 2/3.

Test yourself

The distance from a point to a line calculator finds the distance from a point to a line given by its general equation or by two points, in the plane and in space, and the distance from a point to a plane. It also finds the foot of the perpendicular — the point of the line closest to the given point (the projection of the point onto the plane).

Distance formulas

Problem	Formula
Point M(x₀, y₀) and line Ax + By + C = 0	d = \|Ax₀ + By₀ + C\| / √(A² + B²)
Point and line AB in space	d = \|AM × AB\| / \|AB\|
Point and plane Ax + By + Cz + D = 0	d = \|Ax₀ + By₀ + Cz₀ + D\| / √(A² + B² + C²)

Example 1. M(3, 4), line 3x + 4y − 5 = 0.
d = |9 + 16 − 5| / √(9 + 16) = 20 / 5 = 4.

Example 2. M(1, 2, 3), plane 2x − 2y + z − 4 = 0.
d = |2 − 4 + 3 − 4| / √(4 + 4 + 1) = 3 / 3 = 1.

The space formula uses the cross product: its length is the area of the parallelogram on AM and AB, and the distance is the height of that parallelogram to side AB. If the distance is zero, the point lies on the line (in the plane).