1. Distance Formula

We have a right-angled triangle with hypotenuse length c, as shown:

Recall Pythagoras' Theorem, which tells us the length of the longest side (the hypotenuse) of a right triangle:


We use this to find the distance between any two points (x1, y1) and (x2, y2) on the cartesian (x-y) plane:

The Cartesian Plane

The cartesian plane was named after Rene Descartes.

See more about Descartes in Functions and Graphs.

x y
A (x1, y1)
B (x2, y1)
C (x2, y2)
distance = d

The point B (x2, y1) is at the right angle. We can see that:

  • The distance between the points A(x1, y1) and B(x2, y1) is simply x2x1 and
  • The distance between the points C(x2, y2) and B(x2, y1) is simply y2y1.
x y
A (x1, y1)
B (x2, y1)
C (x2, y2)
x2 − x1
y2 − y1
distance = d

Distance from (x1, y1) to (x2, y2).

Using Pythagoras' Theorem we can develop a formula for the distance d.

Continues below

Distance Formula

The distance between (x1, y1) and (x2, y2) is given by:


Note: Don't worry about which point you choose for (x1, y1) (it can be the first or second point given), because the answer works out the same.

Interactive Graph - Distance Formula

You can explore the concept of distance formula in the following JSXGraph (it's not a fixed image).

Drag either point A or point B to investigate how the distance formula works.

You can move the graph up-down, left-right if you hold down the "Shift" key and then drag the graph.

Sometimes the explanation boxes overlap. It can't be helped (especially on a small screen)!

If you get lost, you can always refresh the page.

Example 1

Find the distance between the points (3, −4) and (5, 7).

Example 2

Find the distance between the points (3, −1) and (−2, 5).

Example 3

What is the distance between (−1, 3) and (−8, −4)?

Example 4

Find k if the distance between (k,0) and (0, 2k) is 10 units.


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