The distance formula

The distance formula helps you calculate how far apart two points in a coordinate system are. To do this, it uses the Pythagorean theorem and its properties. Like this:

Draw a point in two dimensional Cartesian space. Now draw a line connecting that point with the point of origin. Do you see it? If not, draw a line passing through the point you selected that is perpendicular to the horizontal x-axis. You should notice that the line you just drew, together with the part of the x-axis and the line connecting your point to the point of origin, forms a right triangle. The line that connects your point and O is the hypotenuse of said triangle and the other two lines are the legs.

distance formula

Now, you probably remember the Pythagorean theorem from the last lesson. If not, click on the Pythagorean theorem and refresh your memory. If you do, you probably remember that the length of the hypotenuse in a right triangle can be calculated using the formula:

c2 = a2 + b2

This is the basis of the distance formula. The square of the length of the hypotenuse is the sum of squares of the lengths of both legs. The length of a leg of the triangle in a coordinate system is the distance between two points in space that are the endpoints of that particular line.  For clarity, let us call the line along the x-axis side a, and the line parallel to the y axis side b. Since none of the points that make side a do not change their position in relation to the y axis, the distance between them is simply the difference in size of the x-coordinates of the endpoints. Like this:

a = x2 – x1

And that is the length of side a. The length of side b can be calculated in much the same way, only the coordinate that remains fixed is the x-coordinate (instead of the y-coordinate, as it was with side a). That means that the length of side b is:

b = y2 – y1

Now since we know both legs, we can insert these small expressions in our equation and get:

C2 = (x2 – x1)2 + (y2 – y1)2

If you calculate the square root of this equation, you will get what is called the distance formula.

The distance formula can be applied to calculate the distance between any two points in Euclidean space and it will be very useful in many occasions.

If you wish to practice what you learned about the distance formula, please feel free to use the math worksheets below.

The distance formula exams for teachers

Exam Name File Size Downloads Upload date
Integers
The distance formula – Integers – easy 203.7 kB 4075 October 13, 2012
The distance formula – Integers – medium 206 kB 3590 October 13, 2012
The distance formula – Integers – hard 238 kB 2162 October 13, 2012
The distance formula – Integers – very hard 279.4 kB 1613 October 13, 2012
Decimals
The distance formula – Decimals – easy 276.2 kB 1071 October 13, 2012
The distance formula – Decimals – medium 284.3 kB 1063 October 13, 2012
The distance formula – Decimals – hard 292 kB 1123 October 13, 2012
The distance formula – Decimals – very hard 306.7 kB 896 October 13, 2012
Fractions
The distance formula – Fractions – easy 773.4 kB 1014 October 13, 2012
The distance formula – Fractions – medium 797.5 kB 916 October 13, 2012
The distance formula – Fractions – hard 800.8 kB 931 October 13, 2012


The distance formula worksheets for students

Worksheet Name File Size Downloads Upload date
The distance formula – Integers 2.9 MB 3698 October 14, 2012
The distance formula – Decimals 541 kB 1330 October 14, 2012
The distance formula – Fractions 3.6 MB 1061 October 14, 2012
The distance formula – Integers – Line segment 12.8 MB 2257 October 14, 2012