Understanding the Distance Formula

Calculate the distance between two points on a Cartesian plane

Business people looking through picture frame at the road ahead
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The Cartesian plane distance formula determines the distance between two coordinates. You'll use the following formula to determine the distance (d), or length of the line segment, between the given coordinates.

d=√((x1-x2)2+(y1-y2)2)

How the Distance Formula Works

distance formula

Consider a line segment identified by using the coordinates on a Cartesian plane.

To determine the distance between the two coordinates, consider this segment as a segment of a triangle. The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse of the triangle will be the distance between the two points.

Making a Triangle

An illustration of the distance formula on the plane.
Jim.belk/Wikimedia Commons/Public Domain

To clarify, the coordinates x2 and x1 form one side of the triangle; y2 and y1 compose the third side of the triangle. Thus, the segment to be measured forms the hypotenuse and we are able to calculate this distance.

The subscripts refer to the first and second points; it doesn't matter which points you call first or second:

  • x2 and y2 are the x,y coordinates for one point
  • x1 and y1 are the x,y coordinates for the second point
  • d is the distance between the two points