Skip to main content
Search IntMath
Close

1. Distance Formula

We use the Distance Forumala to find the distance between any two points (x1,y1) and (x2,y2) on a cartesian plane.

See also:

The Cartesian Plane

The cartesian plane was named after Rene Descartes.

See more about Descartes in Functions and Graphs.

Let's start with 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:

`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:

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.

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 interactive graph (it's not a fixed image).

Drag either point A (x1, y1) or point C (x2, y2) to investigate how the distance formula works. As you drag the points the graph will automatically calculate the distance.

Length AB = x2x1

Length BC = y2y1

Length

Copyright © www.intmath.com

Example 1

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

Answer

Here, x1 = 3 and y1 = −4; x2 = 5 and y2 = 7

So the distance is given by:

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

`=sqrt((5-3)^2+(7-(-4))^2)`

`=sqrt(4+121)`

`=11.18`

Example 2

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

Answer

This time, x1 = 3 and y1 = −1; x2 = −2 and y2 = 5

So the distance is given by:

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

`=sqrt((-2-3)^2+(5-(-1))^2)`

`=sqrt(25+36)`

`=sqrt61`

`=7.8102`

Example 3

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

Answer

In this example, x1 = −1 and y1 = 3; x2 = −8 and y2 = −4

So the distance is given by:

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

`=sqrt((-8-(-1))^2+(-4-3)^2)`

`=sqrt(49+49)`

`=sqrt98`

`=9.899`

Example 4

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

Answer

This is the situation:

Graph solution - straight line

Applying the distance formula, we have:

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

`=sqrt((2k-0)^2+(0-k)^2)`

`=sqrt(4k^2+k^2)`

`=sqrt(5k^2)`

Now `sqrt(5k^2)=10` so `5k^2=100`, giving:

k2 = 20

so

`k=+-sqrt(20)~~+-4.472`

We obtained 2 solutions, so there are 2 possible outcomes, as follows:

Graph solutions - 2 straight line

Problem Solver

AI Math Calculator Reviews

This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of GPT large language models to parse and generate natural language. This creates math problem solver thats more accurate than ChatGPT, more flexible than a calculator, and faster answers than a human tutor. Learn More.

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.