How this instrument works
The Euclidean distance between two points is the length of the straight line connecting them — found by treating the horizontal and vertical gaps between the points as the two legs of a right triangle, then applying the Pythagorean theorem to find that triangle's hypotenuse, which is exactly the distance being measured.
This page covers the simpler, introductory two-dimensional case: points described by just an x and a y coordinate each. This site's separate Distance calculator extends the identical idea to three dimensions, adding a z coordinate and one more squared term under the same root — the same core idea, just one dimension further.
The formula only ever cares about the DIFFERENCE between each pair of coordinates, never their absolute position — shifting both points by the same amount in any direction leaves the distance between them completely unchanged.
- Enter the first point's x and y coordinates.
- Enter the second point's x and y coordinates.
- Read Distance: the straight-line length between the two points.
- Swap the two points' coordinates to confirm the distance comes out identical either way.
Worked example — (0,0) to (3,4)
The distance between (0,0) and (3,4) is √(3²+4²)=√25=5 — the Pythagorean theorem applied directly, treating the horizontal gap (3) and vertical gap (4) between the two points as the legs of a right triangle whose hypotenuse is the distance itself.
The distance between (2,3) and (7,15) is √(5²+12²)=√169=13, a 5-12-13 right triangle hiding in the coordinate gaps. The distance between (−1,−1) and (2,3) is also √(3²+4²)=5 — negative coordinates work exactly the same way, since only the DIFFERENCE between each pair of coordinates ever enters the calculation.
Questions
What is the Euclidean distance formula?
d = √((x₂−x₁)² + (y₂−y₁)²), the Pythagorean theorem applied to the horizontal and vertical gaps between two points on a flat plane — the straight-line distance connecting them.
How is this different from the Distance calculator on this site?
That page covers three dimensions, adding a z coordinate and a third squared term under the same root; this page covers the simpler two-dimensional introductory case, using just x and y coordinates.
Does it matter which point is entered first?
No — swapping the two points just flips the sign of each coordinate difference, and since both differences are squared before adding, the final distance comes out identical either way.
What if the two points share the same x-coordinate?
The horizontal term drops to zero, and the distance simplifies to just the difference in y-coordinates — the two points sit directly above and below one another, a purely vertical separation.
Why is it called Euclidean distance specifically?
It's named for the geometry of Euclid, the ordinary flat-plane geometry most people learn first — other, non-Euclidean ways of measuring distance exist for different contexts, but this straight-line version is the most familiar and widely used.