Have you ever had a fish tank? Take a look at this dilemma.

Carlos is cleaning out his fish tank, so he filled the bathtub to the rim with water for his fish to swim in while he empties their tank. If the bathtub is 5.5 feet long, 3.3 feet wide, and 2.2 feet deep, how many cubic feet of water can it hold?

**To solve this problem, you will need to understand volume. Pay attention to this Concept and you will know how to solve this dilemma by the end of it.**

### Guidance

**What is volume?**

*Volume***of a solid figure is the measure of how much three-dimensional space it takes up or holds.**

Imagine an aquarium for fish. Its length, width, and height determine how much water the tank will hold. If we fill it with water, the amount of water tells the volume of the tank.

**We measure volume in cubic units, because we are multiplying three dimensions: length, width, and height.**

There are several different ways that we can calculate volume. The first way that we are going to explore is by looking at we can calculate volume using unit cubes.

**What are unit cubes?**

Unit cubes are singular cubes used to represent one unit. When we “fill up” a solid figure with unit cubes, we can see the unit cubes lining up the figure. Then we can count or calculate the number of unit cubes in the solid. The number of unit cubes in the solid figure is the volume of the figure. Let’s examine what this would look like in the following prism.

Now you can see that this prism holds unit cubes. It has three unit cubes lined up for the length, it has two unit cubes along the side for the width and it has four unit cubes lined up for the height. If we count all of these unit cubes, then we can see that we have 24 unit cubes.

**We would write our answer for volume as 24 cubic units. Notice that we use cubic units because we have length times width times height.**

The formula for volume is a simplified version of the method we just learned. As we saw, we used length and width to find the number of unit cubes in the first layer of the figure. This is the same as finding the area of its base. Let’s see how this works.

In this case, the base of the prism is a rectangle. We can use the area formula for rectangles to find the area of the base: \begin{align*}A = lw\end{align*}. This is the same as counting the number of unit cubes in each row and the number of rows. Once we find the area, we simply multiply it by the height to add the rest of the layers. Therefore the formula for the volume of a rectangular prism is

@$\begin{align*}V = Bh\end{align*}@$

**@$\begin{align*}B\end{align*}@$ represents the base area of the prism. Remember, a prism can have a base in the shape of any polygon. Therefore, the formula we need to use to find the area of the base will change. But the process stays the same: we find the area of the face that is the base and then multiply this by the height of the prism. Remember, you will need to figure out the area of the base and then multiply it by the height to figure out the volume.**

Let's try it out.

*Find the volume of the prism below.*

**We simply put the values for the length, width, and height in for the appropriate variables in the formula. Then we solve for @$\begin{align*}V\end{align*}@$, volume.**

**First we find the area of the base. This is the rectangular side on the bottom. Remember, to find the area of a rectangle we multiply the length times the width. This is our first step.**

@$$\begin{align*}A&=lw\\ A&=(16)(9)\\ A&=144 \ sq.cm\end{align*}@$$

**The area of the base is 144 sq. cm.**

**But we are not done yet. We need to figure out the volume, so we will need to take the measurement for the area of the base of the figure and multiply it by the height of the figure. We use the following formula to calculate this measurement.**

@$$\begin{align*}V&=Bh\\ V&=144(4)\\ V&=576 \ cm^3\end{align*}@$$

**The volume of this rectangular prism is @$\begin{align*}576 \ cm^3\end{align*}@$.** *Notice that we used the exponent three to represent the cubic units of the figure. All volume is measured in cubic units, so you will need to use this exponent when working on figuring out the volume of a solid.*

*Now write this formula down in your notebook. Be sure to make a note that you will need to find the area of the base of the figure and that this could be different depending on the figure.*

**We can also use what we have learned to work backwards.** If we know the volume and the area of the base of a prism, then we can figure out the height of the prism which would be the missing dimension. Remember that this is the same thing as working out a puzzle!

*The base of a rectangular prism with a volume of 1,145.52 cubic feet has sides of 17.2 feet and 11.1 feet. What is the height of the prism?*

**First, we need to find the area of the base, @$\begin{align*}B\end{align*}@$. We know this is a rectangular prism, so we use the formula @$\begin{align*}B = lw\end{align*}@$.**

@$$\begin{align*}B &= lw\\ B &= 17.2 (11.1)\\ B &= 190.92 \ ft^2\end{align*}@$$

**We can put this into the formula for @$\begin{align*}B\end{align*}@$. We also have been given the volume of the prism, so we put this in for @$\begin{align*}V\end{align*}@$. Then we solve for @$\begin{align*}h\end{align*}@$, the height.**

@$$\begin{align*}V &= Bh\\ 1,145.52 &= 190.92h\\ 1,145.52 \div 190.92 &= h\\ 6 \ ft &= h\end{align*}@$$

**The height of the prism is 6 feet.**

Find the volume of each prism.

#### Example A

A triangular prism with a @$\begin{align*}b = 12 \ in, \ h = 10 \ in, \ H = 15 \ in\end{align*}@$

**Solution: @$\begin{align*}900 \ in^3\end{align*}@$**

#### Example B

A rectangular prism with a length of 8 m, width of 7 m, height of 3 meters

**Solution: @$\begin{align*}168\end{align*}@$ @$\begin{align*}m^3\end{align*}@$**

#### Example C

A rectangular prism with a length of 10 in, width of 8 in, height of 6 inches

**Solution: @$\begin{align*}480\end{align*}@$ @$\begin{align*}in^3\end{align*}@$**

Now let's go back to the dilemma from the beginning of the Concept.

**First of all, what is the problem asking us to find? We need to find the volume of the bathtub. Is a bathtub a prism or cylinder? It is a rectangular prism, so we’ll need to use the area formula for rectangles to find @$\begin{align*}B\end{align*}@$.**

@$$\begin{align*}B &= lw\\ B &= 5.5 \times 3.3\\ B &= 18.15 \ ft^2\end{align*}@$$

**Now we put this value into the volume formula and solve.**

@$$\begin{align*}V &= Bh\\ V &= 18.15 \times 2.2\\ V &= 39.93 \ ft^3\end{align*}@$$

**Carlos’s bathtub can hold 39.93 square feet of water.**

### Guided Practice

Here is one for you to try on your own.

What is the volume of the prism below?

**Solution**

As we have seen, the volume formula for any prism is @$\begin{align*}V = Bh\end{align*}@$. First we need to find the base area. Take a look at the prism above. The base is a triangle, so this time we need to use the formula for the area of a triangle, @$\begin{align*}\frac{1}{2} bh\end{align*}@$, to find @$\begin{align*}B\end{align*}@$. The height of the triangle, @$\begin{align*}h\end{align*}@$, is indicated by a dashed line. The base of the triangle, @$\begin{align*}b\end{align*}@$, is the side perpendicular to the height. Remember, we use the height and base measurements for the triangular face, not the height measurement for the whole prism. Look carefully at the image!

**Now let’s use the formula for finding the area of a triangle to find the area of base of the triangle.**

@$$\begin{align*}B &= \frac{1}{2} bh\\ B &= \frac{1}{2}(16)(6)\\ B &= 8(6)\\ B &= 48 \ in.^2\end{align*}@$$

**We now have the base area: 48 square inches.**

**Next, we simply multiply it by the height of the prism, according to the volume formula.**

@$$\begin{align*}V &= Bh\\ V &= 48 (10)\\ V &= 480 \ in.^3\end{align*}@$$

The volume of this triangular prism is @$\begin{align*}480 \ in^3\end{align*}@$.

### Video Review

Khan Academy Solid Geometry Volume

### Explore More

Directions: Look at each figure and then answer the questions about each figure.

- Name the figure pictured above.
- What is the length of the figure?
- What is the width of the figure?
- What is the height of the figure?
- What is the volume of the figure?

- What is the name of the figure pictured above?
- What is the shape of the bases?
- What is the shape of the side faces?
- What is the volume of the figure?

- What is the name of the figure pictured?
- What is the volume of the figure?

Directions: Use what you have learned about volume to solve each problem.

- A rectangular prism has a base measuring 16.2 by 14.8 feet. If its volume is 2,877.12 cubic feet, what is its height?
- Kelly is using a rectangular container to fill up a bucket of water. The container is 3.8 inches long, 2.5 inches wide, and 7.2 inches tall. If the bucket holds 1,368 cubic inches of water, how many times will Kelly have to fill the cup in order to fill the bucket?
- True or false. Volume can be used to measure the amount of water in a pool.
- True or false. Surface area and volume measure the same thing.