Lesson 13: The Volume of a Cylinder

Let’s explore cylinders and their volumes.

13.1: A Circle's Dimensions

A circle with the center labeled A. Points B, C, and D lie on the circle, where B is to the right of A, C is to the left of A, and point D is above and to the left of A. A line segment is drawn from A to D and is labeled 4. Another line segment is drawn from B to C such that the segment goes through A.

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 $h$.

    A collection of six cylinder images labeled "A" through "F". Image "A" is the drawing of a cylinder that lies on its bottom base; Image "B" is the drawing of solid, green cylinder tilted to the right; Image "C" is an image of an oatmeal container in the shape of a cylinder; Image "D" is the drawing of a cylinder that lies on its rectangular face; Image "E" is an image of a tanker, 18-wheeler truck which in the shape of a cylinder Image "F" is an image of a farm grain silo that is in the shape of a cylinder. 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.

    A right circular cylinder with a height of 4 and a diameter of 10.
    1. Shade the cylinder’s base.
    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.

A drawing of a cylinder whose radius is 2 and height is 5.

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 $$V=\pi r^2h$$

Practice Problems ▶