Distance between two points calculator

The distance between two points on a plane is the length of the line segment joining them. If you plot the points on a set of coordinate axes, the segment AB is the hypotenuse of a right-angled triangle whose two shorter sides are the differences between the coordinates.

How it works

By Pythagoras’ theorem, the distance between the points A(x₁, y₁) and B(x₂, y₂) is

$$|AB|=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$$

The order of the points does not matter, because the difference is squared, so the minus sign disappears.

For the points A(1, 2) and B(4, 6) we first subtract the coordinates:

$$x_2-x_1=4-1=3,y_2-y_1=6-2=4$$

Substituting into the formula:

$$|AB|=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$$

The points are exactly 5 units apart. The same calculation lies behind the length of the vector $\overrightarrow{AB}$.