Topics
7. Geometry
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United States of America
Grade 7

7.08 Surface area of cylinders

Lesson
Practice

Introduction

Because we have learned how to find the area of circles, and we have an understanding of surface area for prisms and pyramids, we should be able to find the surface area of cylinders.

Surface area of cylinders

Let's see how a cylinder would look if we open it up and view its net.

Exploration

Consider the following questions.

  1. What type of shape did the outside of the cylinder become when it was unfolded?

  2. How many circles are in the net?

  3. Why does the last stage show us that the base circle can roll across the rectangle?

  4. What is the relationship between the circles and the rectangle?

Loading interactive...

Notice that when the curved surface of a cylinder is unfolded, it is a rectangle. By rotating the circle on top of the rectangle, can you see how the circumference of the circle is equal to the length of the rectangular piece?

From this applet we can see that:

  • The surface area of a cylinder is made up of two circles and one rectangle.

  • The radii of both circles is r.

  • The width of the rectangle is h.

  • The length of the rectangle is equal to the circumference of the circle, 2\pi r.

From the applet, we can see that there are three parts to a cylinder's surface area - two circles and a rectangle. We can find the area of these three parts and then add them together to get the total surface area:

\text{Surface area of a cylinder} = \text{Area of } 2 \text{ circular ends} + \text{Area of rectangular piece}

We know that the area of a circle is given by the equation:

\text{Area of a circle}=\pi r^2

Because we have two equal circles we can use the equation

\text{Area of a circle}=2\pi r^2

The equation for the area of a rectangle is given by the equation:L \times WWe now know that the length, L, of the rectangle is the height of the cylinder, h.

By rotating the circle on top of the rectangle, we can see how the circumference of the circle is equal to the width, W, of the rectangular piece. The circumference of a the circle is given by 2\pi r, so we have:

\text{Area of the rectangular piece}=2\pi r \times h

Now that we have the area of all of the parts, we can put them together to get the total surface area.

Examples

Example 1

Consider the following cylinder and it's corresponding net:

A cylinder and its net. The cylinder has a radius of 3 meters and height of 5 meters. The net is composed of 2 identical circles with a radius of 3 meters, and a rectangle with a width of 5 meters.
a

Find the area of one of the circular faces of the cylinder. Use 3.14 for \pi and round your answer to two decimal places.

Worked Solution
Create a strategy

Remember that the area of a circle is A=\pi r^2

Apply the idea
\displaystyle A\displaystyle =\displaystyle 3.14 \times 3^2Substitute the values
\displaystyle =\displaystyle 28.26Evaluate

The area of a circle of the cylinder is 28.26\text{ m}^2.

b

Find the area of the curved face. Use 3.14 for \pi and round your answer to two decimal places.

Worked Solution
Create a strategy

The area of the curved face of the cylinder will be the area of the middle rectangle, which makes up the side of the cylinder.

The area of the rectangle is equal to the product of its length and width.

The width of the rectangle is equal to the given height of the cylinder. What will the length of the rectangle be?

Apply the idea

The length of the rectangle is equal to the circumference of the circular base which is 2 \pi r. Therefore, the surface area of the curved face of a cylinder is 2 \pi r \times h.

The cylinder has a height of 5\text { m}, and a radius of 3 \text { m}.

\displaystyle \text{Curved surface area}\displaystyle =\displaystyle 2 \times 3.14 \times 3 \times 5Substitute the values
\displaystyle =\displaystyle 94.20Evaluate

The curved surface area is 94.20 \text{ m}^2.

c

Find the total surface area of the cylinder. Round your answer to two decimal places.

Worked Solution
Create a strategy

The surface of a cylinder is made up of two circular faces on the top and bottom and a rectangular face that wraps around the curved surface of the cylinder.

Apply the idea

The surface area of the cylinder can be calculated by adding the areas of each part:

\displaystyle \text{Surface area of a cylinder}\displaystyle =\displaystyle \text{Area of } 2 \text{ circular bases} + \text{Area of rectangular piece}
\displaystyle SA\displaystyle =\displaystyle 2\times 28.26\text{ cm}^2 + 94.20 \text{ cm}^2Substitute the values from (a) and (b)
\displaystyle =\displaystyle 150.72 \text{ cm}^2Evaluate

The surface area of the cylinder is 150.72 \text{ cm}^2.

Idea summary

A cylinder is a 3D shape much like a prism with two identical circular bases and a curved surface that joins the two bases together.

The surface area of the cylinder can be calculated by totaling the area of the parts:

\text{Surface area of a cylinder} = \text{Area of } 2 \text{ circular ends} + \text{Area of rectangular piece}

Outcomes

7.G.B.4

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

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