Distance Formula Calculator
Free distance formula calculator: enter two coordinate points to get the distance with every substitution step shown, plus Δx, Δy and the slope.
About the Distance Formula Calculator
The distance between two points (x₁, y₁) and (x₂, y₂) on a coordinate plane comes from the Pythagorean theorem: d = √((x₂ − x₁)² + (y₂ − y₁)²). This free distance formula calculator computes the distance instantly and — unlike a plain calculator — writes out every substitution step, so you can copy the working straight into homework or check your own algebra.
It also reports Δx and Δy (the horizontal and vertical changes) and the slope of the line through the points as a bonus. Coordinate geometry is a staple of matric, FSc, O-level and engineering-entry syllabi in Pakistan and India, and the same formula powers everything from game development to mapping. Everything runs live in your browser with no signup.
How to Use the Distance Formula Calculator
- 1Enter x₁ and y₁ for the first point.
- 2Enter x₂ and y₂ for the second point.
- 3Read the distance, Δx, Δy and the full substituted working.
Frequently Asked Questions
How do I use the distance formula with an example?
For (1, 2) and (4, 6): d = √((4−1)² + (6−2)²) = √(3² + 4²) = √(9 + 16) = √25 = 5. The calculator shows each of these substitution lines so you can follow or copy the working.
Why does the distance formula work?
It is the Pythagorean theorem in disguise. The segment between the points is the hypotenuse of a right triangle whose legs are Δx = x₂ − x₁ and Δy = y₂ − y₁, so d² = Δx² + Δy². That is why the (3, 4) legs in the example give a distance of 5.
Does the order of the points matter?
No. Swapping the points flips the signs of Δx and Δy, but squaring removes the signs: (−3)² = 3² = 9. Distance is always positive (or zero for identical points), whichever point you call first.
Can the distance formula work in 3D?
Yes — add a z term: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). For (0,0,0) and (2, 3, 6): √(4 + 9 + 36) = √49 = 7. This tool handles the 2D case, which covers most coursework.