Introduction

In this lesson, you will find the volume of a rectangular object. To do this, you must find the area of the base and multiply it by the height. You can find the volume of a cylinder in the same way.

This video illustrates the lesson material below. Watching the video is optional.

• Volume of a Right Circular Cylinder (05:54 mins) | Transcript

Volume of a Right Circular Cylinder: $V=\pi r^{2}h$

When you find the volume of a rectangular object, you find the area of the base and multiply it by the height. Do the same thing to find the volume of a cylinder only this time the base is a circle. Find the area of the base $\pi r^2$ and multiply that by the height of the cylinder.

Here is some vocabulary to help with this lesson.

• Radius (r): The distance from the center of the circle to the edge or half of the diameter.
• Height (h): The distance from the bottom to the top of a shape or object.
• Right Angle: This is the same thing as perpendicular, two lines come together at 90 degrees like the corner of a rectangle.
• Right Circular Cylinder: A shape like a tube, the ends (or base) form a circle and the sides are perpendicular to the base. The circular bases will always be parallel for Right Circular Cylinders.

Figure 1

Example 1
The radius of the circular base of a cylinder is 1.5 centimeters and the height is 5 centimeters. What is the volume of the cylinder?

Figure 2

$$\begin{array}{rlr}V& =\pi r^{2}h& \text{Formula for volume of a cylinder}\\ \\ V& =\pi (1.5cm{)}^2(5cm)& \text{Substitute given terms}\\ \\ V& =\pi (2.25c{m}^2)(5cm)& \text{Solve exponents}\\ \\ V& =\pi 11.25c{m}^3& \text{Multiply}\\ \\ V& =(3.14)11.25c{m}^3& \text{Substitute }\pi \\ \\ V& =35.34c{m}^3& \text{Multiplication property}\end{array}$$

Example 2
The radius of the circular base of a cylinder is 2.4 inches and the height is 6 inches. What is the volume of the cylinder? Round to the nearest tenth.

$$\begin{array}{rlr}V& =\pi r^{2}h& \text{Formula for volume of a cylinder}\\ \\ V& =\pi (2.4in{)}^2(6in)& \text{Substitute given terms}\\ \\ V& =\pi (5.76i{n}^2)(6in)& \text{Solve exponents}\\ \\ V& =\pi 34.56i{n}^3& \text{Multiply}\\ \\ V& =(3.14)34.56i{n}^3& \text{Substitute }\pi \\ \\ V& =108.5i{n}^3& \text{Multiplication property}\end{array}$$

Things to Remember

• To find the volume of a cylinder, determine the area of the base and multiply it by the height: $V=\pi r^{2}h$.

Practice Problems

1. A can of food is a right circular cylinder with a radius of 5 cm and a height of 16 cm. Find the volume of the can. Round your answer to the nearest tenth. (
   Solution

   x

   Solution: $1256.6{\text{ cm}}^3$ (when using the pi button on the calculator)
   )
2. A paint can is a right circular cylinder with a radius of 3.5 inches and a height of 7.5 inches. Find the volume of the paint can. Round your answer to the nearest hundredth. (
   Solution

   x

   Solution: $288.63{\text{ in}}^3$ (when using the pi button on the calculator)
   )
3. A water tower is used to pressurize the water supply for the distribution of water in the surrounding area. A particular water tower is in the shape of a right circular cylinder with a radius of 4.25 meters and a height of 7.5 meters. Find the volume of the water tower. Round your answer to the nearest whole number. (
   Solution

   x

   Solution: $426{\text{m}}^3$ (when using the pi button on the calculator)
   Details:
    
   You are finding the volume of a water tower with a radius of 4.25 meters and a height of 7.5 meters, so you can use the formula for the volume of a right cylinder:
   
   $\text{Volume}=\text{π}{\text{ r}}^2\text{h}$
   
   The first thing to do is substitute or replace ‘r’ with 4.25 m and ‘h’ with 7.5 m
   
   $\text{Volume}=\text{π}(4.25\text{m}){}^2(7.5\text{m})$
   
   Next, square $4.25\text{m}$ to get $18.0625{\text{m}}^2$ (this means multiply $4.25\text{m}\times 4.25\text{m}$).
   
   $\text{Volume}=\text{π}(18.0625{\text{m}}^2)(7.5\text{m})$
   
   Then multiply $18.0625{\text{m}}^2$ by $7.5\text{m}$, which gives you $135.46875{\text{m}}^3$. Remember ${\text{m}}^2$ times $\text{m}$ equals ${\text{m}}^3$ and tells you that you are measuring the volume in cubic meters.
   
   $\text{Volume}=\text{π}135.46875{\text{m}}^3$
   
   Since you can multiply in any order, you can rewrite the equation like this, which is an acceptable mathematical answer:
   
   $\text{Volume}=135.47\text{π}{\text{m}}^3$ (Here it is also rounded to the nearest hundredth place for simplicity.)
   
   You can also multiply 135.46875 by $\pi$ to get:
   
   $\text{Volume}=425.5876...{\text{m}}^3$
   
   The volume of the water tower is approximately $426{\text{m}}^3$ when rounded to the nearest whole number.
   )
4. A 55-gallon drum is in the shape of a right circular cylinder with a diameter of 22.5 inches and a height of 33.5 inches. First, find the radius of the drum and then use the radius to find the volume of the drum. Round your answer to the nearest hundredth. (
   Video Solution

   x

   Solution:
   $\text{Radius}=11.25\text{ in}$
   $\text{Volume}=13319.86{\text{ in}}^3$ (when using the pi button on the calculator)
   Details:
   

   
   
   | Transcript)
5. A support column on a building is a right circular cylinder. It has a radius of 1.5 feet and a height of 16 feet. Find the volume of the column. Round your answer to the nearest whole number. (
   Solution

   x

   Solution: $113{\text{ ft}}^3$ (when using the pi button on the calculator)
   Details:
    
   To find the volume of the column you can use the formula:
   
   $\text{Volume}=\text{π}{\text{ r}}^2\text{h}$
   
   The first thing to do is replace the ‘r’ with 1.5 ft and the ‘h’ with 16 ft.
   
   $\text{Volume}=\text{π}(1.5\text{ft}){}^2(16\text{ft})$
   
   Next, square 1.5 ft (this means multiply $1.5\text{ft}\times 1.5\text{ft}$ which equals $2.25{\text{ft}}^2$).
   
   $\text{Volume}=\text{π}(2.25{\text{ ft}}^2)(16\text{ft})$
   
   Then multiply $2.25{\text{ ft}}^2$ by $16\text{ft}$, which equals $36{\text{ft}}^3$. Remember ${\text{ft}}^2$ times $\text{ft}$ equals ${\text{ft}}^3$.
   
   $\text{Volume}=\text{π}36{\text{ft}}^3$
   
   Since you can multiply in any order, you can rewrite the volume like this:
   
   $\text{Volume}=36\text{π}{\text{ft}}^3$
   
   When you multiply 36 by $\pi$ you get:
   
   $\text{Volume}=113.0973...{\text{ft}}^3$
   
   The volume of the column is about $113{\text{ft}}^3$ when rounded to the nearest whole number.
   )
6. A triple-A battery is a right circular cylinder with a radius of 5.25 mm and a height of 44.5 mm. Find the volume of the battery. Round to the nearest tenth. (
   Video Solution

   x

   Solution: $3853.3{\text{ mm}}^3$ (when using the pi button on the calculator)
   Details:
   

   
   
   | Transcript)