How the distance formula works
The distance formula is the Pythagorean theorem applied to coordinates: d = √((x2 − x1)² + (y2 − y1)²). For points (1, 2) and (4, 6), d = √(3² + 4²) = √(9 + 16) = √25 = 5 units. The steepness of that same line comes from the slope calculator.
To see why, draw a right triangle with the segment as its hypotenuse: one leg runs horizontally Δx = x2 − x1, the other vertically Δy = y2 − y1. Pythagoras then gives d² = Δx² + Δy².
How to use this calculator
Enter two points as (x1, y1) and (x2, y2). The calculator squares the horizontal and vertical gaps, adds them, and takes the square root — the straight-line distance. To measure the boundary of a shape instead, use the perimeter calculator.
Keep the coordinates paired correctly: mixing x1 with y2 is the commonest slip. If the two points share an x value the distance is simply the vertical gap.
Worked example
From (1, 2) to (4, 6): Δx = 3, Δy = 4, so d = √(9 + 16) = √25 = 5. This is the classic 3-4-5 right triangle.
Most coordinate pairs give an irrational decimal instead. Note also that this is the straight-line, or Euclidean, distance: if movement is restricted to a city grid you would add Δx and Δy, giving 3 + 4 = 7 rather than 5. The formula is the Pythagorean theorem in disguise: the distance is the hypotenuse of a right triangle, which the hypotenuse calculator finds from the two legs.
Frequently asked questions
- What is the distance formula?
- d = √((x2 − x1)² + (y2 − y1)²) — the square root of the squared horizontal gap plus the squared vertical gap.
- What is the distance between (1, 2) and (4, 6)?
- 5 units, because √((4 − 1)² + (6 − 2)²) = √(9 + 16) = √25 = 5.
- Is the distance formula the same as Pythagoras?
- Yes — it is the Pythagorean theorem where the horizontal and vertical differences are the two legs of a right triangle.
- Can the distance be negative?
- No. Distance is always zero or positive because the differences are squared before the square root.
- Does the order of the two points matter?
- No — swapping the points gives the same answer. Reversing them flips the sign of Δx and Δy, but squaring removes the sign, so d is unchanged. Only the reported Δx and Δy in the badge change sign, which tells you the direction of travel.