Lesson 13: The Volume of a Cylinder

Let’s explore cylinders and their volumes.

13.1: A Circle's Dimensions

Here is a circle. Points $A$, $B$, $C$, and $D$ are drawn, as well as Segments $AD$ and $BC$.

1. What is the area of the circle, in square units? Select all that apply.
1. $4\pi$
2. $\pi 8$
3. $16\pi$
4. $\pi 4^2$
5. approximately 25
6. approximately 50
2. If the area of a circle is $49\pi$ square units, what is its radius? Explain your reasoning.

13.2: Circular Volumes

What is the volume of each figure, in cubic units? Even if you aren’t sure, make a reasonable guess.

1. Figure A: A rectangular prism whose base has an area of 16 square units and whose height is 3 units.
2. Figure B: A cylinder whose base has an area of 16$\pi$ square units and whose height is 1 unit.
3. Figure C: A cylinder whose base has an area of 16$\pi$ square units and whose height is 3 units.

13.3: A Cylinder's Dimensions

1. For cylinders A–D, sketch a radius and the height. Label the radius with an $r$ and the height with an $h$.

Silo, Water Tank Volvo water tank truck in Iraq Copyright Owner: Jum Gordon, N3dling License: Public Domain Via: Pixabay
2. Earlier you learned how to sketch a cylinder. Sketch cylinders for E and F and label each one’s radius and height.

13.4: A Cylinder's Volume

1. Here is a cylinder with height 4 units and diameter 10 units.

2. What is the area of the cylinder’s base? Express your answer in terms of $\pi$.
3. What is the volume of this cylinder? Express your answer in terms of $\pi$.
2. A silo is a cylindrical container that is used on farms to hold large amounts of goods, such as grain. On a particular farm, a silo has a height of 18 feet and diameter of 6 feet. Make a sketch of this silo and label its height and radius. How many cubic feet of grain can this silo hold? Use 3.14 as an approximation for $\pi$.

Summary

We can find the volume of a cylinder with radius $r$ and height $h$ using two ideas we've seen before:

• The volume of a rectangular prism is a result of multiplying the area of its base by its height.
• The base of the cylinder is a circle with radius $r$, so the base area is $\pi r^2$.

Remember that $\pi$ is the number we get when we divide the circumference of any circle by its diameter. The value of $\pi$ is approximately 3.14.

Just like a rectangular prism, the volume of a cylinder is the area of the base times the height. For example, take a cylinder whose radius is 2 cm and whose height is 5 cm.

The base has an area of $4\pi$ cm2 (since $\pi\boldcdot 2^2=4\pi$), so the volume is $20\pi$ cm3 (since $4\pi \boldcdot 5 = 20\pi$). Using 3.14 as an approximation for $\pi$, we can say that the volume of the cylinder is approximately 62.8 cm3.

In general, the base of a cylinder with radius $r$ units has area $\pi r^2$ square units. If the height is $h$ units, then the volume $V$ in cubic units is $$V=\pi r^2h$$