How to Find the Distance Between Two Points

Finding the Distance Between Two Points

The distance between two points with Cartesian coordinates (x1, y1) and (x2, y2) can be found using the formula:

The image below shows what we mean by the distance between the points at (x1, y1) and (x2, y2):

x1, y1, x2 and y2 are symbols that represent the x-coordinates and y-coordinates of the points. In real questions, the Cartesian coordinates will have numbers, for example (1, 1) and (5, 4).

How to Find the Distance Between Two Points

Finding the distance between two points is easy.

Question

Find the distance between the points with Cartesian coordinates (1, 1) and (5, 4).

Step-by-Step:

1

Start with the formula:

$$Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Don't forget: √ means square root

and 2 means squared: (x2 − x1)2 = (x2 − x1) × (x2 − x1)

and 2 means squared: (y2 − y1)2 = (y2 − y1) × (y2 − y1)

2

Find x1, y1, x2 and y2 from the Cartesian coordinates given in the question.

In our example, the Cartesian coordinates of the points are (1, 1) and (5, 4). They are represented in the formula by (x1, y1) and (x2, y2).

(x1, y1) = (1, 1) ∴ x1 = 1, y1 = 1

(x2, y2) = (5, 4) ∴ x2 = 5, y2 = 4

3

Substitute x1, y1, x2 and y2 into the formula.

$$Distance = \sqrt{(5 - 1)^2 + (4 - 1)^2}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{4^2 + 3^2}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{(4 \times 4) + (3 \times 3)}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{16 + 9}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{25}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = 5$$

Answer:

The distance between the points with Cartesian coordinates (1, 1) and (5, 4) is 5.

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The slider below shows another real example of how to find the distance between two points.

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See Also

What are Cartesian coordinates? What is the x-coordinate? What is the y-coordinate? What is a square root? What is a square number? What is Pythagoras' Theorem? What is a right triangle?