# How do Calculate Distance and Examples

Ashley Kelton, Jeff Calareso
• Author
Ashley Kelton

Ashley Kelton has taught Middle School and High School Math classes for over 15 years. During her 15 years of teaching, she has taught Algebra, Geometry, and AP Calculus. She has a Bachelor’s of Science in Elementary Education from Southern Illinois University and a Master’s of Science in Mathematics Education from Southern Illinois University. She also has a Professional Teaching Certificate in Math grades 6-12 and Elementary Education.

• Instructor
Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Discover different distance equations. Learn how to calculate distance using coordinates and using the distance rate time formula. See various distance formula examples. Understand how to find miles per hour using distance formula word problems. Updated: 10/18/2021

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## Distance Equations

Distance is a measurement. Distance refers to the length between two points or between two objects. For example, you might need the distance between your hometown and another city so you can calculate how much gas you may need for a trip. Likewise, you may need the distance between two walls to determine how much paint you need to paint a room. Sometimes distance, however, is not referred to as distance in a problem, but perhaps the length, width, height, or how far are used in place of the word distance.

You can calculate the distance in a couple of ways. One way is to use the rate of what you are calculating and multiply it by the time it takes. This distance formula is written as {eq}d=rt {/eq}. The other way to calculate distance is to use the coordinate plane. This distance equation is written as {eq}d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} {/eq}. In this equation, you will need to have the (x, y) coordinate pair for two points to calculate the distance. An error occurred trying to load this video.

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• 0:03 Solve for Distance
• 2:12 Solve for Rate
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• 3:45 Distance Plus Distance
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## How to Calculate Distance Using Coordinates

A coordinate provides a location of a point in the coordinate plane. Each coordinate has two numbers in it separated by a comma and placed in parentheses like the following example: {eq}(1, 2) {/eq}. The first number represents the x value of the coordinate, and the second number represents the y value of the coordinate. To calculate the distance between the two points shown in the example above, you will need the distance equation (also referred to as distance formula in geometry) {eq}d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} {/eq} and you will need the coordinates of two points. The two points in the example above are {eq}(1,2) {/eq} and {eq}(-1,-6) {/eq}. In the formula, you will notice that there are subscripts 1 and 2. These subscripts are just telling you that you have a first and second x value and a first and second y value. Another way to think about it is that you have two points. You have a first point and a second point. It does not matter which point you call the first and which you call the second. So, {eq}(1,2) {/eq} can be considered the first or second point when using the distance equation. And, {eq}(-1,-6) {/eq} can be considered the first or second point when using the distance equation.

Using the example mentioned above, here is how the distance equation would work if you use {eq}(1,2) {/eq} as the first point.

{eq}d=\sqrt{(1+1)^2+(2+6)^2} {/eq}

{eq}d=\sqrt{2^2+8^2} {/eq}

{eq}d=\sqrt{4+64}=\sqrt{68}\approx8.24621 {/eq}

Here is how the distance equation would work if you use {eq}(-1,-6) {/eq} as the first point.

{eq}d=\sqrt{(-1-1)^2+(-6-2)^2} {/eq}

{eq}d=\sqrt{(-2)^2+(-8)^2} {/eq}

{eq}d=\sqrt{4+64}=\sqrt{68}\approx8.24621 {/eq}

You can see that it does not matter which point you identify as the first and identify as the second, as you will get the same answer for both.

So far, the distance equation has been discussed in regards to a two-dimensional plane. If however, you would like to use the distance equation in three-dimensional problems, you will have to use the following equation: {eq}d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} {/eq}.

### Distance Formula Examples

#### Example 1

Given the following image, calculate the distance between the two points. The two points in the image are {eq}(4,3) {/eq} and {eq}(-2, 8) {/eq}.

Plug these two coordinates into the distance formula.

{eq}d=\sqrt{(-2-4)^2+(8-3)^2} {/eq}

{eq}d=\sqrt{(-6)^2+(5)^2} {/eq}

{eq}d=\sqrt{36+25} {/eq}

{eq}d=\sqrt{61}\approx7.8102 {/eq}

The distance between the two coordinates is approximately {eq}7.8102 {/eq} units.

#### Example 2

In this example, we will look at how to use the distance formula in a three-dimensional situation. The coordinates in this example are {eq}(1,2,3) {/eq} and {eq}(-4,5,-7) {/eq}.

Plug these two coordinates into the distance formula.

{eq}d=\sqrt{(-4-1)^2+(5-2)^2+(-7-3)^2} {/eq}

{eq}d=\sqrt{(-5)^2+(3)^2+(-10)^2} {/eq}

{eq}d=\sqrt{25+9+100} {/eq}

{eq}d=\sqrt{134}\approx11.5758 {/eq}

The distance between these two coordinates is approximately {eq}11.5758 {/eq} units.

## Distance Rate Time Formula

When given a real-world situation that involves rates and time, there is another way to calculate the distance. Use the distance rate time formula :{eq}d=rt {/eq}.

In this formula, you must make sure that you are using the same units of measure for the rate and the time. For instance, if you have a rate of {eq}50 {/eq} miles per day, then your time must also be in days.

### Distance Formula Word Problems

#### Example 3

Denise has been riding her bike for a few weeks now. She is building her stamina to participate in a local event in which the route is a total distance of {eq}16 {/eq} miles. She knows that she can complete {eq}5 {/eq} miles in one hour. The event is scheduled to take place over {eq}4 {/eq} hours. Denise wants to know how far she would travel in {eq}4 {/eq} hours if she bikes at her current rate to make sure she can finish the event in the allotted time. Calculate her distance over the {eq}4 {/eq} hours. Use the following formula.

{eq}d=rt {/eq}

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#### How do you calculate distance traveled?

You calculate distance traveled by using the formula d=rt. You will need to know the rate at which you are traveling and the total time you traveled. You can then multiply these two numbers together to determine the distance traveled.

#### How do you find the distance formula?

The distance formula is found by solving the Pythagorean Theorem for c. The Pytahgorean Theorem is a^2+b^2=c^2. Once you have solved this for c, you will have the distance formula.

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