# Distance Formula & Calculator

With the use of the distance calculator, we are able find the distance between two points (x₁,y₁) and (x₂,y₂) on a Cartesian coordinate system. It uses the common distance formula and will compute this distance when you simply enter the coordinates. As you will see in the next section, the distance equation will show how to find the distance between two points and is closely related to the Pythagorean theorem.

## The distance formula

Suppose we have coordinates `(3,5) and (9,15)`

and we want to calculate the distance. The distance formula is `√(x₂ - x₁)² + (y₂ - y₁)²`

, which relates to the Pythagorean theorem, which is `a² + b² = c²`

, where `a`

and `b`

are legs of a right triangle and `c`

is the hypotenuse. Suppose (x₁,y₁) and (x₂,y₂)are coordinates of the endpoints of the hypotenuse. Then `(x₂ - x₁)²`

in the distance equation corresponds to `a²`

and `(y₂ - y₁)²`

corresponds to `b²`

. Since `c = √a² + b²`

, you can see how it is just an extension of the Pythagorean theorem.

## How to find distance between two points

The steps for how to find the distance between two points are as follows:

- Input the values into the formula:
`√(x₂ - x₁)² + (y₂ - y₁)²`

. - Subtract values in parentheses.
- Square both quantities in parentheses.
- Add the results.
- Take the square root.
- Use the distance equation to check results.

Working the example by hand, we get `√(9-3)²+(15-5)²`

= `√36 + 100`

= `√136`

, which is approximately equal to 11.66. Note that when taking the square root you will get a positive and negative value, but since we are dealing with distance, we are only concerned with the positive result.

## Other considerations related to the distance calculator

An application of the distance calculator is that is can be used together with the gas calculator.

Suppose you are traveling from city A to city B and the only reference is city C, with route A to B perpendicular to route B to C. We can determine the distance from A to B. From this the gas calculator will determine fuel cost, fuel used and cost per person while traveling.

Another common calculation used with line segments is calculating the midpoint. The is important when determining segment bisectors in geometric proofs and constructions. Calculating the midpoint involves coordinates (x₁,y₁) and (x₂,y₂), as does the distance calculator and is done simply with the midpoint calculator

The more complex version of the distance formula can be used to find the distance between coordinates in a three-dimensional coordinate system. The formula involves coordinates (x₁,y₁,z₁) and (x₂,y₂.z₂) and is √(x₂ - x₁)² + (y₂ - y₁)² +(z₂ - z₁)².