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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

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Instructor:
*Kimberly Osborn*

Volume is simply the amount of space that a substance (or object) takes up in a container. In this lesson, you will specifically explore the equation and application for the volume of a box.

A few years back, I moved into a new home with a big backyard and an empty swimming pool in the shape of a box. After having the pool cleaned and resurfaced, I was stuck with the dilemma of trying to figure out how much water I needed to fill it. This is where our lesson topic comes into play.

Now, before I begin explaining the definition and equation for the volume of my swimming pool, let's first look at a diagram and see how we would label this diagram to organize our thoughts.

Below you will see a box (or in more formal language, rectangular prism) with the following sides labeled, *l*, *w*, and *h*. Each of these labels correspond with the box's:

- Length (
*l*) - Width (
*w*) - Height (
*h*)

However, the length, width, and height of the box are all based on the perspective of how you are looking at the box. When looking at any box, I would recommend picking one corner (I prefer the right bottom corner as shown in the above image) and labeling the three sides that shoot out for that corner.

- The length follows the horizontal line on the front of your box.
- The height follows the vertical line on the front of your box.
- And the width follows the line from the front to the back of the box.

Looking back at the example given in the introduction of this lesson, we were tasked with finding the amount of water needed to fill my box shaped swimming pool. This is the classic definition of volume.

**Volume** is simply the amount of space that a substance (or object) takes up in a container.

In the case of my swimming pool, the water is our substance and the pool itself is our container.

To find the volume (typically noted with a capital *V*) all we need to do is multiply the length by the width and the height.

The equation for the volume of a box is then:

**Special Case:**

When working with a cube (all sides are the same length), your length, width, and height will all be the same. Therefore, the equation for the volume of a cube will be:

V = side x side x side (or side^3)

We know that every time we are working with measurements, we must have units. For example, the units for area of a rectangle is units^2, or square units. These could be inches^2, feet^2, yards^2, etc.

This is the exact same concept for volume. Keep in mind however, that instead of just multiplying length x height to get our area, we have an additional side that we are multiplying, the width, to get our volume. This means that our units would then be units^3 or cubic units. Again, these could be inches^3, feet^3, yards^3, etc.

In the case of our example, my swimming pool has a length of 20 feet, height of 12 feet, and width of eight feet. Using our equation and understanding of units, how much water do I need to fill my swimming pool?

To solve this example, let's first write our equation for volume and then plug in our given numbers. After multiplying through, our final answer is 1,920. Since we are working with feet, our units would then be feet^3, or cubic feet. So, I would need 1,920 cubic feet of water to fill my entire swimming pool.

In this lesson, we learned that volume is the amount of space that a substance (or object) fills in a container. To find the volume of a box, or rectangular prism, we simply need to multiply the length of the box by its width and then height. As the final step of your solution, don't forget your cubic units!

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11 in chapter 18 of the course:

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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

- Perimeter of Triangles and Rectangles 8:54
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