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):

`c=sqrt(a^2+b^2)`

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:

`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2`

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!

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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