The Distance Formula

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The Distance Formula (page 1 of 2)

The Distance Formula is a variant of the Pythagorean Theorem that you used back in geometry. Here's how we get from the one to the other:

Suppose you're given the two points (–2, 1) and (1, 5), and they want you to find out how far apart they are. The points look like this:

plotted points

You can draw in the lines that form a right-angled triangle, using these points as two of the corners:

right triangle

It's easy to find the lengths of the horizontal and vertical sides of the right triangle: just subtract the x-values and the y-values:

lengths

Then use the Pythagorean Theorem to find the length of the third side (which is the hypotenuse of the right triangle):

c² = a² + b²

c = 5

This format always holds true. Given two points, you can always plot them, draw the right triangle, and then find the length of the hypotenuse. The length of the hypotenuse is the distance between the two points. Since this format always works, it can be turned into a formula:

Distance Formula: Given the two points (x₁, y₁) and (x₂, y₂), the distance between these points is given by the formula:

Don't let the subscripts scare you. They only indicate that there is a "first" point and a "second" point; that is, that you have two points. Whichever one you call "first" or "second" is up to you. The distance will be the same, regardless.

Find the distance between the points (–2, –3) and (–4, 4).

I just plug the coordinates into the Distance Formula:

d = sqrt(53)

Then the distance is sqrt(53), or about 7.28, rounded to two decimal places.

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Cite this article as:

Stapel, Elizabeth. "The Distance Formula." Purplemath. Available from
http://www.purplemath.com/modules/distform.htm. Accessed

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