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How Many Jelly Beans?

Have you ever had to guess how many jelly beans will fit into a jar? Knowing how to estimate the capacity of the jar and the volume of a single jelly bean can help you come up with a good guess. 

Volume is the amount of space an object occupies. One way to find the volume of an object is to submerge in water, and measure the volume of water that it displaces.

The volume of water the object displaces is equal to the volume of the object.

Now if you don't happen to have a big bucket of water handy, or perhaps the object would not do well submerged in water, then there are formulas that we can use to find the volume instead.

Sources: Jar - Zerbor/iStock/Getty Images; Jelly Beans - Joe_Potato/iStock/Getty Images; 
Other Jar - vinap/iStock/Getty Images; Rock - Yevhenii Dubinko/iStock/Getty Images 

Lesson Goals

• Determine the formula for finding the volume of a prism.
• Convert between units of volume.
• Relate volume and capacity.
• Calculate the volume of prisms and composite solids.

Try This!

The space inside the following glass jar has dimensions as marked on the diagram and a single jelly bean has a volume of \(3.5\) cm\(^3\). 

Estimate how many jelly beans the jar can hold.

Think about this problem, then move on to the next part of the lesson.

What Is Volume?

Definition of a Prism

You can create a 3D solid by sweeping a 2D figure through space. For example, consider the following square. Picture what would happen if you move this square straight up. You would call the distance that you move the square the height. By doing this, you have created a right prism.

Identifying Bases of Prisms

We can think of a prism as being formed by sweeping one base through space to the other base. Since the bases are important, we should be able to identify the bases of any prism.

Check Your Understanding 1

Question

Identify the bases of the prism.

Answer

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/QzVECmxM

Feedback

The two selected faces are identical in shape and the other faces are all shaped like parallelograms. That manes the selected faces are bases.

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Volume of a Prism

The volume of a prism is the amount of space it occupies.

Consider a rectangular prism with a square base. You might recall that you can calculate the volume of this prism using the formula \(V  =l\times w\times h\).

If we add in some dashed lines to help us visualize the faces of this prism that we cannot see, then what we notice is that the volume is really the area of the base multiplied by the height of the prism.

And so the volume is equal to the area of the base times the height. 

\(V = A_{rectangle} \times h\)

The volume of any prism can be calculated in this way.

We measure volume in cubic units such as mm\(^3\), cm\(^3\), or m\(^3\). And we do this because we are multiplying area, which is measured in square units, by a third dimension, height.

Check Your Understanding 2

Question

Calculate the volume, in cm\(^3\), of the rectangular prism below.

Answer

\(1235\) cm\(^3\)

Solution

To calculate the area of this rectangular prism, we multiply the area of the base by the height of the prism.

\(\begin{align*} V & = A_{base} \times h \\ & = 95 \times 13 \\ & = 1235 \end{align*}\)

Therefore, the rectangular prism has a volume of \(1235\) cm\(^3\).

Example 1

Solution

We can calculate the volume of this prism by multiplying the area of the base, which is the triangular face, by the height of the prism.

\(\begin{align*} V &= A_{base} \times h \\ &\; = \dfrac{1}{2}(10\times 8.5) \times 20 \\ &\; = \dfrac{1}{2}(85) \times 20 \\ &\; = 42.5 \times 20 \\ &\; = 850 \end{align*}\)

Therefore, this prism has a volume of \(850\) cm\(^3\).

Check Your Understanding 3

Question

Calculate the volume, in cm\(^3\), of the cube below.

Answer

\(512\) cm\(^3\)

Feedback

To calculate the area of this cube, we multiply the area of the base by the height of the prism.

\(\begin{align*} V & = A_{base} \times h \\ & = (8\times 8) \times 8 \\ & = 64 \times 8 \\ & = 512 \end{align*}\)

Therefore, the cube has a volume of \(512\) cm\(^3\).

Converting Between Units of Volume

Volume in m\(^3\) and cm\(^3\)

Consider a cube with edges that are \(1\) m in length. To calculate the volume of this cube, we would multiply the area of the base by the height and find that the volume is \(1\) m\(^3\).

But since \(1\) m is equal to \(100\) cm, we can take the same cube and label the edges as \(100\) cm instead of \(1\) m.

Since we have calculated the same volume just using different units, the two volumes must be equal.

What we can conclude is that \(1\) m\(^3 = 1~000~000\) cm\(^3\).

Example 2

Solution

Since we want to convert centimeters cubed to meters cubed, we know that we're going to want to use the fact that \(1~000~000\) cm\(^3\) \( = 1\) m\(^3\). Now we need to set up the information that we're given so that this information is useful.

To do this, we notice that \(2~500~000\) is equal to \(2.5 \times 1~000~000\). Why did we do this? Well, because now, we have the \(1\) million that we need to use our conversion. And so we can replace \(1\) million centimetres cubed with \(1\) metre cubed as follows:

\[\begin{align*} 2~500~000 \text{ cm}^3 &\; \class{timed in4}{= 2.5 \times 1~000~000 \text{ cm}^3} \\ &\; \class{timed in5}{= 2.5 \times 1 \text{ m}^3} \\ &\; \class{timed in6}{= 2.5 \text{ m}^3} \end{align*}\]

Therefore, the volume of the prism is \(2.5\) m\(^3\).

 

Check Your Understanding 4

Question — Part A

Which of the following is correct?

1. \(1 \text{ cm}^3 = 1000 \text{ mm}^3\)
2. \(1 \text{ cm}^3 = 10 \text{ mm}^3\)
3. \(1 \text{ cm}^3 = 10~000 \text{ mm}^3\)
4. \(1 \text{ cm}^3 = 100 \text{ mm}^3\)

Answer — Part A

2. \(1 \text{ cm}^3 = 10 \text{ mm}^3\)

Feedback — Part A

Therefore, \(1 \text{ cm}^3 = 1000 \text{ mm}^3\).

Question — Part B

A prism has a volume of \(4.3\) cm\(^3\). What is the volume measure in mm\(^3\)? Use your answer from part a) to help you.

Answer — Part B

\(4300\) mm\(^3\)

Feedback — Part B

Using the fact that \(1 \text{ cm}^3 = 1000 \text{ mm}^3\),

\(\begin{align*} 4.3 \text{ cm}^3 & = 4.3 \times 1 \text{ cm}^3 \\ & = 4.3 \times 1000 \text{ mm}^3 \\ & = 4300 \text{ mm}^3 \end{align*}\)

Therefore, the volume of the prism is \(4300\) mm\(^3\).

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Measuring Volume Using Litres

Another common unit used to measure volume is the litre. 

\(1\) L is equal in volume to a cube with side lengths of \(10\) cm.

\(\Large 1 \text{ L} = (10 \text{ cm})^3 \; \class{timed in3}{= 1000 \text{ cm}^3}\)

There are alternative ways that we can express this conversion. One common one is the following:

\(\Large 1 \text{ mL} = 1 ~\text{cm}^3\)

But why do we need these conversions? Well, the volume of liquids and gases are usually measured in millilitres and litres. Yet their containers have length, width, and height that are measured using millimetres, centimetres, and metres. Being able to relate the two methods of measurement is helpful.

Example 3

Solution

Since we want to convert litres to centimeters cubed, we want to use the fact that \(1\) L \( = 1000\) cm\(^3\). Now again, we want to set up the information that we're given so that this conversion is useful.

To do this, we notice that \(1.2\) L is equal to \(1.2 \times 1\) L.

We now have the \(1\) L that we need to use our conversion. And so we're going to replace \(1\) L with \(1000\) cm\(^3\), and we evaluate to get \(1200\) cm\(^3\). We summarize as follows:

\[\begin{align*} 1.2 \text{ L} &\; \class{timed in4}{= 1.2 \times 1 \text{ L}} \\ & \; \class{timed in5}{= 1.2 \times 1000 \text{ cm}^3} \\ & \; \class{timed in6}{= 1200 \text{ cm}^3} \end{align*}\]

Therefore, \(1.2\) L is equal to \(1200\) cm\(^3\).

How might knowing the volume as \(1200\) cm\(^3\) instead of \(1.2\) L be useful?

Well, one reason is that we can now use the volume in cubic centimetres to come up with some possible dimensions for the juice container.

For example, the following containers all have a volume of \(1200\) cm\(^3\). Can you come up with some others?

What dimensions might be the better choice for the juice carton?

Capacity

What Is Capacity?

Consider the following box.

This box is not a prism because it has hollow space inside and is therefore not a 3D solid.

We use boxes and other containers that are hollow on the inside to hold things. 

Now using our knowledge of prisms, we are able to calculate the capacity of containers, such as this box.

Example 4

Solution

\(\begin{align*} V &= A_{base} \times h \\ &\; \class{timed in3}{= (16 \times 29) \times 9} \\ &\; \class{timed in4}{= 464 \times 9} \\ &\; \class{timed in4}{= 4176 } \end{align*}\)

Since we know the volume of the space inside the container, the capacity of the container is \(4176\) cm\(^3\).

Example 5

Solution

\(\begin{align*} V &= A_{base} \times h \\ &\; \class{timed in3}{= (6 \times 3) \times 10} \\ &\; \class{timed in4}{= 18 \times 10} \\ &\; \class{timed in4}{= 180} \end{align*}\)

Therefore, the capacity of the juice box is \(180\) cm\(^3\).

But since we would fill the juice box with liquid, it makes more sense to express the capacity of the juice box in millilitres.

Since \(1\) mL \(= 1\) cm\(^3\), the capacity of the juice box is \(180\) mL.

Check Your Understanding 5

Question

A jelly mold is in the shape of a rectangular prism. 

Its inside dimensions are \(6\) cm by \(6\) cm by \(2\) cm.

What is the capacity of the jelly mold?

Answer

\(72\) mL

Feedback

First, we find the volume of the jelly mold. We will assume the base has dimensions \(6\) cm by \(6\) cm.

\(\begin{align*} V &  = A_{base} \times h \\ & = (6\times 6) \times 2 \\ & = 36 \times 2 \\ & = 72 \end{align*}\)

Therefore, the capacity of the jelly mold is \(72\) cm\(^3\).

It makes more sense to express the capacity of the jelly mold in mililitres. Since \(1\) ml \(= 1 \) cm\(^3\), the capacity of the jelly mold is \(72\) mL.

Composite Solids

Example 6

A composite solid is made up of two or more prisms joined together.

Solution

Just like we did when finding the area of composite shapes, we find the volume of composite solids by decomposing the solid into simpler solids for which we have simple volume formulas.

Notice that this solid is composed of a smaller rectangular prism where the length is \(3\) m, width is \(2\) m, and height is \(4\) m.

The larger rectangular prism has length \(3\) m, width \(3\) m, and height \(6\) m.

The volume of the total solid is equal to the volume of the small prism plus the volume of a large prism.

The volume of the smaller prism is equal to the area of its base times its height.

\(\begin{align*} V_{small} &\; \class{timed in6}{= (3\times 2) \times 4 }\\ &\; \class{timed in7}{= 6 \times 4} \\ &\; \class{timed in7}{= 24} \end{align*}\)

Similarly, the volume of the larger prism is equal to the area of its base times its height.

\(\begin{align*} V_{large} &\; \class{timed in10}{= (3\times 3) \times 6 }\\ &\; \class{timed in11}{= 9 \times 6} \\ &\; \class{timed in11}{= 54} \end{align*}\)

We now look at the equation for the total volume and evaluate:

\[\begin{align*} V_{total} &= V_{small} + V_{large} \\ &\; \class{timed in8}{= 24 \;+\;} \class{timed in13}{54} \\ &\; \class{timed in14}{= 78} \end{align*}\]

The volume of the composite solid is \(78\) cm\(^3\).

Check Your Understanding 6

Question

Find the volume of the following composite solid.

Answer

\(1512\) m\(^3\)

Feedback

Try dividing the shape as shown and calculating the volume of each part separately.

\(\begin{align*}V_{total} & = V_{top} + V_{bottom} \\ & = 1152+360 \\ & = 1512 \end{align*}\)

Thus, the volume is \(1512\) m\(^3\).

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/BDK497G7

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Try This Problem Revisited

We can also use our knowledge of composite solids to solve problems involving capacity. To see this, let's revisit to try this problem.

Solution

Looking at this jar notice that it can be thought of as a composition of two identical trapezoidal prisms. We can find the capacity of the jar by finding the volume of these two prisms. 

The capacity of the jar is \(1600\) cm\(^3\).

Now that we know the capacity of the jar, our second step is to determine how many jelly beans will fit inside. We can do this by dividing the capacity of the jar by the volume of one jelly bean. If each jelly bean has a volume of \(3.5\) cm\(^3\) and

\[1600 \div 3.5 \approx 457.14\]

we would estimate that the jar could hold about \(457\) jelly beans. 

Now, an important question to consider if you were actually submitting a guess is would you actually expect the jar to hold \(457\) jelly beans? Think about why the number in the jar should be slightly less than that.

Try This Problem Revisited — Continued

Alternate Solution

Now, the try this problem is interesting. Because you can actually approach finding the capacity of the jar in a different way. Alternatively, you can view the base of the jar as a composite shape because it's a hexagon, which we can see as two identical trapezoids.

The capacity of the jar is \(1600\) cm\(^3\). 

Notice that even though we thought about this problem in a different way, the algebra actually works out to be quite similar.

Check Your Understanding 7

Question

The space inside a cardboard box is in the shape of a rectangular prism. It has a length of \(35\) cm, a width of \(35\) cm, and a height of \(24\) cm. A single golf ball has a volume of approximately \(39.4\) cm\(^3\).

We want to estimate how many golf balls the cardboard box can hold. Which of the following estimates is most reasonable?

1. \(373\)
2. \(1492\)
3. \(2238\)
4. \(746\)

Answer

4. \(746\)

Feedback

Step 1: Calculate the capacity of the cardboard box.

\(\begin{align*} V & = 35 \times 35 \times 24 \\ & = 29~400 \end{align*}\)

Thus, the capacity of the cardboard box is \(29~400\) cm\(^3\).

Step 2: Estimate the number of golf balls.

Each golf ball has a volume of approximately \(39.4\) cm\(^3\), so

\(29~400 \div 39.4 \approx 746\)

We would estimate that the cardboard box could hold about \(746\) golf balls.

Take It With You

A cooler has an advertised capacity of \(17\) L.

You measure the dimensions of the cooler and determine that the capacity should be about \(40\) L.

Explain why the advertised capacity is different than what you calculated.