Distance Between Points Calculator

About Distance Between Points Calculator

The distance formula is derived from the Pythagorean theorem. It gives the straight-line distance between two points.

How to use this calculator

1. Enter the first point as \((x_1, y_1)\) and the second point as \((x_2, y_2)\).
2. Check that both points are written in the same coordinate plane.
3. Use the formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
4. Read the result as the straight-line distance between the two points.

The formula explained

The formula computes the length of the segment connecting \( (x_1, y_1) \) and \( (x_2, y_2) \) using the horizontal and vertical differences. It comes from the Pythagorean theorem.

• \(d\) = the distance between the two points
• \(x_1\) = the x-coordinate of the first point
• \(y_1\) = the y-coordinate of the first point
• \(x_2\) = the x-coordinate of the second point
• \(y_2\) = the y-coordinate of the second point

Step by step method

1. Subtract the x-coordinates to find the horizontal change, and subtract the y-coordinates to find the vertical change.
2. Square both differences so they are positive, then add the results.
3. Take the square root of the sum to get the distance.

Worked example

Problem. Find the distance between \((2, 3)\) and \((8, 15)\).

1. Compute the differences, \(x_2 - x_1 = 8 - 2 = 6\), and \(y_2 - y_1 = 15 - 3 = 12\).
2. Square and add, \(6^2 + 12^2 = 36 + 144 = 180\).
3. Take the square root, so \(d = \sqrt{180} = 6\sqrt{5}\).

Tips and common mistakes

• Be careful with subtraction, because \(x_2 - x_1\) and \(y_2 - y_1\) can be negative before squaring, but the squares make them positive.
• If the points share the same x-coordinate or y-coordinate, the distance is just the vertical or horizontal difference.

Frequently asked questions