The Distance Formula

We all know that the distance between two points of the plane, with the coordinates of Cartesian P (x1, y1) and Q (x2, y2), is provided by the following formula:

PQ = \sqrt{(x2-x1)^{2} + (y2-y1)^{2}}

distance formula

The formula of distance is just the theorem of Pythagoras. For the evaluation of distance PQ between point P (x1, y1) and Q (x2, y2), first, make the right angle triangle which has the segment of PQ as the hypotenuse.

If the lengths of all the sides are p and q, then by the theorem of Pythagoras –

(PQ)2 = (PR)2 + (QR)2

For solving the distance PQ, we have –

PQ = (PR)2 + (QR)2

Since PR is the distance horizontally, it is just the difference between the coordinates of X – |(x2 – x1)|. Similarly, QR is the distance vertically |(y2 – y1). Since we square these distance and squares are non-negative, we need not worry about all these value signs.


Example 1: Find out the distance between the points P and Q.

In the above example, we need –

P(x1, y1) = (-2, 0)
Q(x2, y2) = (3, 8)


PQ = \sqrt{(3 – (-2))^{2} + (8 - 0)^{2}} = 74

Example 2: Find out the distance between the points A and B.

A = (1,1) = (x1, y1)
B = (2,2) = (x2, y2)

AB = \sqrt{(2-1)^{2} + (2-1)^{2}} = \sqrt{2}

Example 3: Find the distance between points E and F.

E= (1,2,3) = (x1, y1, z1)
F = (4,5,6) = (x2, y2, z2)

EF = \sqrt{(4-1)^{2} + (5-2)^{2}+(6-3)^{2}} = \sqrt{27}