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: \(V = \pi r^2h\)
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{align*}V&=\pi r^2h &\color{navy}\small\text{Formula for volume of a cylinder}\\\\V&=\pi(1.5cm)^2(5cm) &\color{navy}\small\text{Substitute given terms}\\\\V&=\pi(2.25cm^2)(5cm) &\color{navy}\small\text{Solve exponents}\\\\
V&=\pi11.25cm^3 &\color{navy}\small\text{Multiply}\\\\V&=(3.14)11.25cm^3 &\color{navy}\small\text{Substitute \(\pi\)}\\\\V&= 35.34cm^3 &\color{navy}\small\text{Multiplication property}\\\end{align*}
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{align*}V&=\pi r^2h &\color{navy}\small\text{Formula for volume of a cylinder}\\\\V&=\pi(2.4in)^2(6in) &\color{navy}\small\text{Substitute given terms}\\\\V&=\pi(5.76in^2)(6in) &\color{navy}\small\text{Solve exponents}\\\\V&=\pi34.56in^3 &\color{navy}\small\text{Multiply}\\\\V&=(3.14)34.56in^3 &\color{navy}\small\text{Substitute \(\pi\)}\\\\V&= 108.5in^3 &\color{navy}\small\text{Multiplication property}\\\end{align*}
Things to Remember
- To find the volume of a cylinder, determine the area of the base and multiply it by the height: \(V = \pi r^2h\).
Practice Problems
- 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. (
- 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. (
- 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. (
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 \(π\) 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. - 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. (
- 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. (
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 \(π\) 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. - 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. (