**Cavalieri’s principle** states that if two solids lying between parallel planes have equal heights and all cross sections at equal distances from their bases have equal areas, then the solids have equal volumes. Why does this make sense?

### Cylinders

Below is a rectangular prism and a cylinder. Note that the height of each solid is the same.

In each case, the area of the base is \begin{align*}\pi r^2\end{align*}. In fact, the area of **any** cross section taken parallel to the base is \begin{align*}\pi r^2\end{align*}. Because these solids have the same height and the same cross sectional areas at every level, the solids have the same volume due to **Cavalieri's principle**.

The volume of the prism is:

\begin{align*}V=A_{Base} \cdot h=\pi r^2 h\end{align*}

Therefore, the volume of the cylinder is:

\begin{align*}V=\pi r^2 h\end{align*}

This should make sense because a cylinder is essentially a circular prism. The area of its base is \begin{align*}\pi r^2\end{align*} and its height is \begin{align*}h\end{align*}, so its volume is \begin{align*}\pi r^2 h\end{align*}.

#### The two cylinders below have the same radius and the same height. Do they have the same volume?

Yes, due to Cavalieri's principle. Even though these two cylinders are different, because they have the same height and base (and because every parallel cross section is congruent to the base), their volumes will be the same. The “slanted” cylinder is called an **oblique cylinder**.

#### Find the volume of the cylinders from the previous problem.

The volume of each cylinder is \begin{align*}V=\pi r^2 h=\pi (2^2) (4)=16 \pi \ in^3\end{align*}.

#### One cup of water has a volume of approximately \begin{align*}14.44 \ in^3\end{align*}. How many cups of water will the cylinders in the first problem hold?

The volume of each cylinder is \begin{align*}16 \pi \ in^3 \approx 50.2655 \ in^3\end{align*}. Since each cup of water has a volume of \begin{align*}14.44 \ in^3\end{align*}, each cylinder will hold \begin{align*}\frac{50.2655}{14.44} \approx 3.5 \ \text{cups of water}\end{align*}.

**Examples**

**Example 1**

Earlier, you were given a problem about Cavalieri's principle.

**Cavalieri's principle** states that if two solids lying between parallel planes have equal heights and all cross sections at equal distances from their bases have equal areas, then the solids have equal volumes.

One way to understand Cavalieri's principle is to imagine a stack of books. Each stack of books below is made up of 15 books. The volume of each stack is the same because the books in each stack are the same. Each stack of books has the same height, and the areas are the same at each cross section that is parallel to the base. Even though the second stack of books is slanted, the volumes are the same.

#### Example 2

Are the volumes of the two cylinders below the same?

No. The height of the oblique cylinder will be less than its slant height of 4 inches. Because the overall height of the two cylinders is not the same, the volumes will be different. *Remember that when calculating the volume, the height you use must be perpendicular to the base.*

#### Example 3

A cylinder is removed from the center of a larger cylinder as shown below:

The radius of the cylinder that was removed is 3 inches. The radius of the large cylinder is 6 inches. The height of the solid is 12 inches. What is the volume of the solid that remains?

The volume of the original large cylinder is \begin{align*}\pi r^2 h=\pi (6^2)(12)=432 \pi \ in^3\end{align*}. The volume of the cylinder that was removed is \begin{align*}\pi r^2 h=\pi (3^2)(12)=108 \pi \ in^3\end{align*}. The volume of the remaining solid is \begin{align*}432 \pi - 108 \pi=324 \pi \ in^3\end{align*}.

#### Example 4

How many cups of water will the solid from #3 hold?

Recall that a cup of water has a volume of approximately \begin{align*}14.44 \ in^3\end{align*}. The volume of the solid from #2 is \begin{align*}324 \pi \approx 1017.876 \ in^3\end{align*}. It will hold \begin{align*}\frac{1017.876}{14.44} \approx 70.5 \ \text{cups of water}\end{align*}. One gallon of water is 16 cups, so this solid will hold approximately 4.4 gallons of water.

### Review

1. Explain Cavalieri's principle in your own words.

2. Explain why the volume of a cylinder with radius \begin{align*}r\end{align*} and height \begin{align*}h\end{align*} is \begin{align*}\pi r^2 h\end{align*}.

3. Explain how the volume of a cylinder relates to the volume of a prism.

Find the volume of each cylinder with the given dimensions.

4.

5.

6. A cylinder with a base diameter of 15 inches and a height of 12 inches.

7. A cylinder with a base diameter of 8 centimeters and a height of 2 centimeters.

8. Find the radius of the base of a cylinder with a volume of \begin{align*}471.24 \ in^3\end{align*} and a height of 6 inches.

9. Find the radius of the base of a cylinder with a volume of \begin{align*}1357.17 \ cm^3\end{align*} if the height of the cylinder is twice the length of the radius.

10. Find the height of a cylinder with a base area of \begin{align*}25 \pi \ in^2\end{align*} if the volume of the cylinder is \begin{align*}300 \pi \ in^3\end{align*}.

11. The label on a can of juice is missing. You want to know how many cups are in the can of juice. You measure the diameter of the base of the can and find that it is 5 inches. You measure the height of the can and find that it is 8 inches. If \begin{align*}14.44 \ in^3\end{align*} is about 1 cup of liquid, how many cups of juice are in the can?

12. A cylinder has been removed from the center of another cylinder. The volume of the remaining solid is \begin{align*}240 \pi \ in^3\end{align*}. If the height of the solid is 3 inches and the radius of the cylinder that was removed is 8 inches, what is the radius of the larger cylinder?

13. How much liquid will the solid from #12 hold if one cup of liquid has a volume of approximately \begin{align*}14.44 \ in^3\end{align*}?

1 cubic centimeter (\begin{align*}cm^3\end{align*} or cc) will hold 1 milliliter (mL) of liquid. Approximately how much liquid will each cylinder hold in liters (1 L = 1000 mL)?

14.

15.

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**Answers for Review Problems**

To see the Review answers, open PDF file and look for section 9.1.