Distance Formula Calculator

Distance formula for two points

The distance formula gives the straight-line distance between two points in the coordinate plane. It comes from the Pythagorean theorem: the horizontal change and vertical change form the legs of a right triangle, and the segment between the points is the hypotenuse. OpenStax explains this connection in its distance and midpoint formulas section.

Worked example

The calculator also shows distance squared, which is useful when comparing lengths without taking a square root.

Distance formula vs. slope vs. midpoint

These three coordinate geometry tools answer different questions. Distance gives the length of a segment, slope gives steepness and direction, and midpoint gives the halfway coordinate. A strong coordinate geometry page should show all three because many homework problems combine them.

Common questions

• What is the distance formula?
  The distance formula is √((x2-x1)^2 + (y2-y1)^2). It finds the straight-line distance between two points.
• Does the order of the points matter?
  No. Reversing the two points changes the signs of Δx and Δy, but squaring them gives the same distance.
• Can distance be negative?
  No. Distance is a length, so it is always zero or positive.
• What does distance squared mean?
  Distance squared is (x2-x1)^2 + (y2-y1)^2 before taking the square root.
• Is the distance formula the same as the Pythagorean theorem?
  It is an application of the Pythagorean theorem on a coordinate plane.
• What if both points are the same?
  The distance is 0 because there is no separation between the points.
• Can I use the distance formula in 3D?
  Yes, but the 3D version adds a z-coordinate: √((x2-x1)^2+(y2-y1)^2+(z2-z1)^2).
• Why do we square the coordinate differences?
  Squaring removes negative signs and measures the horizontal and vertical changes as positive lengths.