Euclidean distance is probably harder to pronounce than it is to calculate. Euclidean distance refers to the distance between two points. These points can be in different dimensional space and are represented by different forms of coordinates. In one-dimensional space, the points are just on a straight number line. In two-dimensional space, the coordinates are given as points on the x- and y-axes, and in three-dimensional space, x-, y- and z-axes are used. Finding the Euclidean distance between points depends on the particular dimensional space in which they are found.

## One-Dimensional

Subtract one point on the number line from another; the order of the subtraction doesn't matter. For example, one number is 8 and the other is -3. Subtracting 8 from -3 equals -11.

Calculate the absolute value of the difference. To calculate the absolute value, square the number. For this example, -11 squared equals 121.

Calculate the square root of that number to finish calculating the absolute value. For this example, the square root of 121 is 11. The distance between the two points is 11.

## Two-Dimensional

Subtract the x- and y-coordinates of the first point from the x- and y-coordinates of the second point. For example, the coordinates of the first point are (2, 4) and the coordinates of the second point are (-3, 8). Subtracting the first x-coordinate of 2 from the second x-coordinate of -3 results in -5. Subtracting the first y-coordinate of 4 from the second y-coordinate of 8 equals 4.

Square the difference of the x-coordinates and also square the difference of the y-coordinates. For this example, the difference of the x-coordinates is -5, and -5 squared is 25, and the difference of the y-coordinates is 4, and 4 squared is 16.

Add the squares together, and then take the square root of that sum to find the distance. For this example, 25 added to 16 is 41, and the square root of 41 is 6.403. (This is the Pythagorean Theorem at work; you are finding the value of the hypotenuse that runs from the total length expressed in x by the total width expressed in y.)

## Three-Dimensional

Subtract the x-, y- and z-coordinates of the first point from the x-, y- and z-coordinates of the second point. For example, the points are (3, 6, 5) and (7, -5, 1). Subtracting the first point's x-coordinate from the second point's x-coordinate results in 7 minus 3 equals 4. Subtracting the first point's y-coordinate from the second point's y-coordinate results in -5 minus 6 equals -11. Subtracting the first point's z-coordinate from the second point's z-coordinate results in 1 minus 5 equals -4.

Square each of the differences of the coordinates. The square of the x-coordinates' difference of 4 equals 16. The square of the y-coordinates' difference of -11 equals 121. The square of the z-coordinates' difference of -4 equals 16.

Add the three squares together, and then calculate the square root of the sum to find the distance. For this example, 16 added to 121 added to 16 equals 153, and the square root of 153 is 12.369.

#### References

- "Geometry: From Euclid to Knots"; Sahl Stahl; 2003
- "Geometry For Dummies"; Mark Ryan; 2008