Knowing how to calculate distance between two coordinates has many practical applications in science and construction. To find the distance between two points on a 2-dimensional grid, you need to know the x- and y-coordinates of each point. To find the distance between two points in 3-dimensional space, you need to know the z-coordinates of the points as well.

The distance formula is used to handle this job and is straightforward: Take the difference between the X-values and the difference between the Y-values, add the squares of these, and take the square root of the sum to find the straight-line distance, as in the distance between two points on Google maps over the ground rather than on a winding road or waterway.

## Distance in Two Dimensions

Calculate the positive difference between the x-coordinates and call this number X. The x-coordinates are the first numbers in each set of coordinates. For example, if the two points have coordinates (-3, 7) and (1, 2), then the difference between -3 and 1 is 4, and so X = 4.

Calculate the positive difference between the y-coordinates and call this number Y. The y-coordinates are the second numbers in each set of coordinates. For example, if the two points have coordinates (-3, 7) and (1, 2), then the difference between 7 and 2 is 5, and so Y = 5.

Use the formula D^{2} = X^{2} + Y^{2} to find the squared distance between two points. For example, if X = 4 and Y = 5, then D^{2} = 4^{2} + 5^{2} = 41. Thus, the square of the distance between the coordinates is 41.

Take the square root of D^{2} to find D, the actual distance between the two points. For example, if D^{2} = 41, then D = 6.403, and so the distance between (-3, 7) and (1, 2) is 6.403.

## Distance in Three Dimensions

Calculate the positive difference between the z-coordinates and call this number Z. The z-coordinates are the third numbers in each set of coordinates. For example, suppose two points in three-dimensional space have coordinates (-3, 7, 10) and (1, 2, 0). The difference between 10 and 0 is 10, and so Z = 10.

Use the formula D^{2} = X^{2} + Y^{2} + Z^{2} to find the squared distance between two points in three-dimensional space. For example, if X = 4, Y = 5, and Z = 10, then D^{2} = 4^{2} + 5^{2}+ 10^{2} = 141. Thus, the square of the distance between the coordinates is 141.

Take the square root of D^{2} to find D, the actual distance between the two points. For example, if D^{2} = 141, then D = 11.874, and so the distance between (-3, 7, 10) and (1, 2, 0) is 11.87.

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