Use cylinder nets to calculate surface area

Ontario Grade 7 Mathematics

Spatial Sense

Find the area of the rectangle and both circular ends

Imagine cutting a closed cylinder-shaped container and laying its outside flat. The flat pattern is called a net. It has one rectangle and two matching circles. Surface area is the total area of all the outside faces. In this lesson, you will use a cylinder net to find that total. We will use centimetres and square centimetres.

What you will learn

Recognize the rectangle and two circles in a closed cylinder net.
Find the rectangle’s area from its side lengths or use a labelled area.
Add the areas of all three faces to find surface area.
State a surface area using square units.

1. Picture the net

A closed cylinder has two flat circular ends and one curved side. When the curved side is unrolled, it becomes a rectangle. The two ends become two matching circles.

Think of a paper label wrapped around a can. If you cut it along the side and lay it flat, the label becomes a rectangle. One side of this rectangle matches the cylinder’s height. The other side matches the distance around a circular end. This distance is called the circumference.

A net shows the faces of a 3-D object laid flat. To find the cylinder’s surface area, find the area of the rectangle and both circles, then add them.

A closed cylinder net has one rectangle and two matching circles.
The rectangle represents the curved side.
Surface area is the total area of all three faces.

2. Find the area of each face

A rectangle’s area is found by multiplying its side lengths. For example, a rectangle that is

5

cm by

3

cm has an area of

15

cm².

The rectangle in a cylinder net has one side equal to the cylinder’s height. Its other side is the distance around a circular end. If both rectangle side lengths are given, multiply them. If its area is already labelled, use that area.

A circle’s area may be labelled on the net. Use the labelled area for each circle. Since the two ends match, their areas are equal. Include both of them.

Area uses square units, such as square centimetres. Do not mix a length in centimetres with an area in square centimetres.

S=R+C_1+C_2

Multiply the two side lengths to find a rectangle’s area.
Include both circular ends.
Add areas, not side lengths.

3. Follow a calculation plan

First, identify the rectangle and both circles. Read each label and decide whether it gives a length or an area.

Next, find the rectangle’s area. Multiply its two side lengths if they are given. If the rectangle’s area is labelled, use that value instead.

Then find the combined area of the two circles. Add their labelled areas. If one circle’s area is given and the two ends match, add that area twice.

Finally, add the rectangle’s area to the two circle areas. The result is the cylinder’s surface area. Keep the answer in square units.

S=R+2C

Use the net’s labels carefully.
Find or read the area of every face.
Add the rectangle and both circles.

4. Check your answer

Count the faces again: one rectangle and two circles. This helps make sure you have included every face of the closed cylinder.

Check that you found areas before adding. If the rectangle is labelled with side lengths, multiply those lengths. If the circles are labelled with areas, use both areas.

Check the units. Multiplying two lengths gives a square-unit area. Adding areas keeps the answer in square units. If the information does not let you find a face’s area, ask for the missing information instead of guessing.

A closed cylinder has three faces in its net.
Do not add lengths to areas.
Write the final answer in square units.

Worked example

Use the labelled areas in a net

A cylinder net has a rectangle measuring

18

cm by

7

cm. Each circular end has an area of

24

cm². Find the surface area.

Schematic cylinder net

Schematic net for the stated measurements. The rectangle is 18 cm by 7 cm, and each circular end has an area of 24 cm². The drawing is not to scale.

Find the rectangle area
The rectangle is $18$ cm by $7$ cm. Multiply the side lengths to find the area of the curved-side face.
$18\times7=126\text{ cm}^2$
Add the circular ends
There are two matching circular ends. Each has an area of $24$ cm², so add that area twice.
$24+24=48\text{ cm}^2$
Add all face areas
Add the rectangle area and the combined area of both circles. All the values are in square centimetres.
$126+48=174\text{ cm}^2$

Answer: The cylinder’s surface area is

174

cm².

Check: The total includes the rectangle and both circular ends.

Worked example

Add the three face areas

A cylinder net labels the rectangle’s area as

95

cm². The two circular ends each have an area of

31

cm². Find the surface area.

Schematic cylinder net with labelled areas

Schematic net showing a rectangle with area 95 cm² and two circles with area 31 cm² each. The drawing is not to scale.

Read the face areas
The net gives the rectangle area and the area of each circular end. These are all the face areas needed.
$95\text{ cm}^2,\quad31\text{ cm}^2,\quad31\text{ cm}^2$
Add the circles
Add the areas of the two circular ends to find their combined area.
$31+31=62\text{ cm}^2$
Find the surface area
Add the combined circle area to the rectangle area.
$95+62=157\text{ cm}^2$

Answer: The surface area is

157

cm².

Check: The calculation includes the rectangle and both circular ends.

Worked example

Reason from the total area

A closed cylinder has a surface area of

210

cm². Its net shows a rectangle with an area of

126

cm². The two circular ends have equal areas. What is the area of each circular end?

Schematic net with a known total

Schematic net: the rectangle area is 126 cm², and the two equal circle areas together make up the rest of the 210 cm² total. The drawing is not to scale.

Find the area of both circles together
The total includes the rectangle and both circles. Subtract the rectangle’s area from the total to find the combined area of the circles.
$210-126=84\text{ cm}^2$
Find the area of one circle
The two circles have equal areas. Share their combined area equally between them.
$84\div2=42\text{ cm}^2$

Answer: Each circular end has an area of

42

cm².

Check: The areas add back to the total:

126+42+42=210

cm².

Common mistakes and how to avoid them

Counting only one circular end.

Correction: A closed cylinder has two circular ends. Include both areas.

Using a rectangle’s side length as its area.

Correction: Multiply the rectangle’s two side lengths, or use its labelled area.

Adding lengths and areas together.

Correction: Find each face’s area first. Then add the areas.

Writing the answer in centimetres instead of square centimetres.

Correction: Surface area is measured in square units, such as cm².

Lesson summary

A closed cylinder net has one rectangle and two circles.
Find the area of each face using the information in the net.
Add the rectangle area and both circular areas.
State surface area in square units.

Check your understanding

Question 1

A net has a rectangle with area

40

cm² and two circles with area

12

cm² each. What is the surface area?

$52$ cm²
$64$ cm²
$76$ cm²
$24$ cm²

Show answer and explanation

76

cm²

The two circles total

24

cm². Add that to the rectangle area:

40+24=64

cm². The correct answer is

64

cm².

Question 2

A cylinder net shows a rectangle measuring

9

cm by

6

cm. What is the rectangle’s area?

$15$ cm²
$54$ cm²
$30$ cm²
$54$ cm

Show answer and explanation

54

cm²

Multiply the side lengths:

9\times6=54

. The area is

54

cm².

Question 3

A closed cylinder has surface area

145

cm². Its rectangle area is

85

cm², and the two circular ends have equal areas. What is the area of each circle?

$30$ cm²
$60$ cm²
$42.5$ cm²
$230$ cm²

Show answer and explanation

30

cm²

The circles together have area

145-85=60

cm². Divide by two because they are equal: each is

30

cm².

Key terms

Net: A flat pattern made by unfolding the faces of a 3-D object.
Surface area: The total area of all the outside faces of a 3-D object.
Circumference: The distance around a circle. In a cylinder net, it matches one side of the rectangle.
Area: A measure of how much flat surface a shape covers.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation E2.6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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E2.6 · Use cylinder nets to calculate surface area

Ontario Grade 7 Mathematics

Find the area of the rectangle and both circular ends

What you will learn

1. Picture the net

2. Find the area of each face

3. Follow a calculation plan

4. Check your answer

Use the labelled areas in a net

Add the three face areas

Reason from the total area

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 7 Math

About this lesson and its review