Distance Formula

How Far Apart Are Two Points?

On a number line, the distance between two points is easy: subtract and take the size. But on a plane, points have both an $x$ and a $y$, so a single subtraction is not enough.

We need a way to measure the straight-line distance between $(x_1, y_1)$ and $(x_2, y_2)$.

It Is Just Pythagoras

Here is the trick. The horizontal gap and the vertical gap between the two points form the two legs of a right triangle. The straight-line distance is the hypotenuse.

• The horizontal leg has length $\Delta x = x_2 - x_1$
• The vertical leg has length $\Delta y = y_2 - y_1$
• The distance $d$ is the hypotenuse

Pythagoras says $d^2 = (\Delta x)^2 + (\Delta y)^2$. Take the square root and you have the formula.

The Formula

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

That is the distance formula, and it is nothing more than the Pythagorean theorem written in coordinates.

Example

Find the distance between $(1, 2)$ and $(4, 6)$.

Order Does Not Matter

You might worry about which point comes first. It never matters, because the differences are squared:

$(x_2 - x_1)^2 = (x_1 - x_2)^2$

A negative gap and a positive gap square to the same thing. Subtract in whichever order you like.

Example With Negatives

Find the distance between $(-2, 3)$ and $(1, -1)$.

The squaring quietly handles every sign for you.