Distance between two parallel lines

The distance between two parallel lines in the plane is the minimum distance between any two points.

Formula and proofEdit

Because the lines are parallel, the perpendicular distance between them is a constant, so it does not matter which point is chosen to measure the distance. Given the equations of two non-vertical parallel lines


the distance between the two lines is the distance between the two intersection points of these lines with the perpendicular line


This distance can be found by first solving the linear systems




to get the coordinates of the intersection points. The solutions to the linear systems are the points




The distance between the points is


which reduces to


When the lines are given by


the distance between them can be expressed as


See alsoEdit


  • Abstand In: Schülerduden – Mathematik II. Bibliographisches Institut & F. A. Brockhaus, 2004, ISBN 3-411-04275-3, pp. 17-19 (German)
  • Hardt Krämer, Rolf Höwelmann, Ingo Klemisch: Analytische Geometrie und Lineare Akgebra. Diesterweg, 1988, ISBN 3-425-05301-9, p. 298 (German)

External linksEdit