Why does the distance formula use a square root?

It comes from solving the Pythagorean theorem for the hypotenuse.

Can distance be negative?

No. Distance is always nonnegative.

What if the two points share an x-value?

The line is vertical, so the slope is undefined while the distance and midpoint are still defined.

Geometry

Distance Formula Calculator

Find the distance, midpoint, and slope between two points with the distance formula and step-by-step work.

Find distance between two points

x1y1x2y2

Answer

d = 5

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
$d = 3^{2} + 4^{2} = 5$
Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = (2.5, 4)
Slope m = (y2 - y1)/(x2 - x1) = 1.333333333

How to use the distance formula calculator

The distance formula is the Pythagorean theorem applied to two points on the coordinate plane. This page also reports the midpoint and slope so you get the full two-point toolkit at once.

Enter two coordinate points.
Subtract x-values and y-values.
Square and add the differences, then take the square root for distance.
Average the coordinates for the midpoint and divide rise by run for the slope.

Formula or method

d = sqrt((x2 - x1)^2 + (y2 - y1)^2); midpoint M = ((x1 + x2)/2, (y1 + y2)/2); slope m = (y2 - y1)/(x2 - x1).

Worked examples

Distance between points

Problem: (1, 2) and (4, 6)

Solution: d = sqrt(3^2 + 4^2) = 5.

Midpoint and slope

Problem: (1, 2) and (4, 6)

Solution: M = (2.5, 4); slope m = 4/3 ≈ 1.33.