Finding the Distance Between Two Points

Being able to find the distance between two points is a very useful "real-life" skill. It is required in many, many careers and tasks. e.g. civil engineering, digital media, military, mapping. Even though, in our computerized world, the calculation is most often done for us, being able to calculate this distance manually helps understanding and reduces the chance of mistakes.

If required, you should review the guidance on graphing x-y coordinates and/ or review Pythagoras' Theorem for help with calculating the lengths of the sides of right-angled triangles.

When calculating the distance between two points on a coordinate system we are applying Pythagoras' Theorem. As shown in the example below, we do this by joining the two points with a straight line and then drawing a right-angled triangle using that straight line as the hypotenuse and aligning the other two sides with the x-axis and y-axis.

We calculate the length of the two sides aligned with the x-axis and y-axis by finding the difference in the x and in the y coordinate values. Using Pythagoras' Theorem we then take the square of these sides, add them, then take the square root which gives the length of the hypotenuse which is the distance between the two points.

Example: Distance Between Two Points

Find the lengths of the short sides of the triangle by finding the differences in the x and in the y coordinates.

9 - 2 = 7
8 - 3 = 5

Square then add these

72 + 52 = 49 + 25 = 74

Take the square root to find the distance between the points.

distance =  74  = 8.602

example showing two points on an x-y grid as being the hypotenuse on a right triangle

Formula For the Distance Between Two Points

The formula below expresses the above description for calculating the distance in algebraic terms. In other words, it says the same thing using symbols and variables.

   
distance = √  (x2 - x1)2 + (y2 - y1)2

 

triangle with hypotenuse between points x1, y1 and x2, y2

The formula can sometimes seem to over-complicate what is essentially just finding the difference in the x and in the y coordinates and then applying Pythagoras' Theorem. It does not matter which coordinate is (x1, y1) and which is (x2, y2) as long as the x and y coordinate values that you subtract are from the same point. .i.e. do not mix them up.

Negative Coordinates

Finding the distance between two points can be just a little harder when one of more negative value is involved. However, as long as the coordinate values are not mixed up and as long as the rules for adding and subtracting negative integers are followed, the same procedure can be followed.

Example With Negative Values

Problem: Find the distance between point (-5,3) and point (4,5) shown on the grid below.

line between points  -5,3 and 4,5

Solution:

Applying the formula

   
distance = √  (x2 - x1)2 + (y2 - y1)2

to find the distance between point (-5,3) and point (4,5) we get

   
distance = √  (-5 - 4)2 + (3-(-5))2
   
distance = √  (-9)2 + (8)2
   
distance = √  145
   
distance =  12.042

Sketch The Problem

If you are presented with just two sets of coordinates and need to find the distance between them then you might find that a quick sketch of the two points on an approximate coordinate grid will help. This helps check that you have the correct differences and also helps check that the answer is reasonable.

Example

Problem: What is the distance between point (-3,-6) and point (8,-2)?

sketch showing two points as vertices on a right triangle

Solution:

Having sketched the two points and the axes we can form right triangle with the two short sides being of lengths 4 and 11. To find the distance we add the squares of both of these and take the square root.

   
distance = √  (4)2 + (11)2
   
distance =  11.705

 

Worksheet

Try this Calculating the Distance Between Two Points Worksheet to practice with questions similar to those above. Note this is a 4-page worksheet and there is an option to show the answers.

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