Grade by Grade
E. Spatial Sense · Spatial Sense: 3D Objects, Transformations, Circles, Surface Area and Volume

Nets and Surface Area of Cylinders

E2.6About 50 minutes
Learning goals
  • I can draw the net of a cylinder: two circles and a rectangle.
  • I can explain why the length of the rectangle equals the circumference of the base.
  • I can find the surface area of a cylinder by adding the areas of its parts.
  • I can solve problems about labels, open containers and wrapping.

Words to know

net
A flat pattern that folds up to make a 3D object.
surface area
The total area of all the surfaces of a 3D object.
lateral surface
The curved side of a cylinder. Unrolled, it is a rectangle.
open cylinder
A cylinder with one base missing, like a cup or a can with no lid.

Remember from Grade 6

In Grade 6 you found the surface area of prisms and pyramids by unfolding them into nets and adding the areas of the faces. A cylinder has a curved side, so its net needs a new idea.

Let's start with a soup can

Carefully peel the paper label off a can of soup (with the can still sealed). Lay the label flat. What shape is it? A rectangle! The curved side of a cylinder unrolls into a rectangle. The top and bottom of the can are circles.

SOUPpeellabellength = circumference = 2 × π × rwidth = height of the can (h)
Peel the label off a can and it lies flat as a rectangle. Its length wraps once around the can, so it equals the circumference; its width is the can's height.

The net of a cylinder

A closed cylinder unfolds into 2 circles and 1 rectangle.

  • The rectangle's width is the height of the cylinder, h.
  • The rectangle's length wraps exactly once around the circle, so it equals the circumference: 2 × π × r.

Surface area = 2 circles + rectangle

SA = 2 × (π × r × r) + (2 × π × r) × h

3 cm10 cm18.84 cmr = 3 cm, h = 10 cmrectangle length= circumference= 2 × 3.14 × 3= 18.84 cm
The net of a cylinder with radius 3 cm and height 10 cm, drawn to scale (5 px for each cm). The rectangle is 18.84 cm long, the same as the circumference of each circle, and 10 cm wide, the height of the cylinder.
Worked example

Example 1: A closed cylinder

Find the surface area of a cylinder with radius 3 cm and height 10 cm.

  1. Two circles: one circle is 3.14 × 3 × 3 = 28.26 cm². Two: 56.52 cm².
  2. Rectangle: length = circumference = 2 × 3.14 × 3 = 18.84 cm. Area = 18.84 × 10 = 188.4 cm².
  3. Total: 56.52 + 188.4 = 244.92 cm².

About 244.92 cm².

Worked example

Example 2: A can label

A can has a diameter of 8 cm and a height of 11 cm. The label covers the whole curved side but not the top or bottom. What is the area of the label?

  1. The label is the rectangle only.
  2. Length = circumference = 3.14 × 8 = 25.12 cm.
  3. Area = 25.12 × 11 = 276.32 cm².

About 276.32 cm².

Worked example

Example 3: An open cylinder

A round planter has no top. Its radius is 5 cm and its height is 12 cm. How much material is needed to make it?

  1. Only one circle (the bottom): 3.14 × 5 × 5 = 78.5 cm².
  2. Rectangle: 2 × 3.14 × 5 = 31.4 cm long. 31.4 × 12 = 376.8 cm².
  3. Total: 78.5 + 376.8 = 455.3 cm².

About 455.3 cm².

Think before you add

  • Is it closed (2 circles), open (1 circle) or a tube or label (0 circles)?
  • Did they give you the radius or the diameter? The circle area needs the radius.
  • Estimate: π is about 3. For Example 1: 2 × 3 × 9 = 54 and 6 × 3 × 10 = 180, total about 234. Close to 244.92 ✓
Hands-on activity

Unroll a tube

You need: an empty paper towel or toilet paper tube, a ruler, scissors, paper and tape. Ask a grown-up to help with scissors.

  1. Measure the diameter and length (height) of the tube.
  2. Predict the area of the curved surface: circumference × height.
  3. Cut the tube straight along its length (if it has a spiral seam, cut straight anyway) and flatten it into a rectangle (or close to one). Measure its length and width. Is the length close to the circumference you calculated?
  4. Make your own cylinder: trace two circles that fit the ends, cut them out, and tape them on. Calculate the total surface area of your closed cylinder.

What to look for: the flattened tube's length is about 3.14 × the diameter.

Just for fun

A soup can is a master of disguise. Peel off the label and the curved surface turns into a plain, flat rectangle. Ta-da!

Your turn

1

What shapes make up the net of a closed cylinder?

2

A cylinder has r = 2 cm and h = 5 cm. What are the length and the area of the rectangle in its net?

3

Same cylinder (r = 2 cm, h = 5 cm). What is its total surface area?

4

Find the surface area of a closed cylinder with diameter 10 cm and height 4 cm.

5

A can has r = 4 cm and h = 12 cm. What is the area of a label that covers its curved side?

6

An open cup has r = 1 cm and h = 10 cm. How much material is used?

More practice: warm-up

1

The width of the rectangle in a cylinder's net equals the cylinder's ______.

2

The length of the rectangle equals the ______ of the base.

3

What is the area of one base of a cylinder with r = 1 cm?

4

What is the circumference of a base with r = 10 cm?

5

How many circles are in the net of a can with its lid on?

More practice: on your own

1

Find the surface area of a closed cylinder with r = 5 cm and h = 20 cm.

2

A disc is 8 cm across and 2 cm thick. Find its surface area.

3

A paper towel tube has a diameter of 4 cm and a length of 28 cm. What is the area of the cardboard (the tube is open at both ends)?

4

A closed water tank is a cylinder with r = 1 m and h = 2 m. How many square metres of paint cover its whole outside?

5

Jo found the surface area of a closed cylinder with d = 6 cm and h = 10 cm. She used 6 as the radius for the circles. Find the correct answer.

Challenge

1

Which closed cylinder needs more material: A with r = 2 cm, h = 8 cm, or B with r = 4 cm, h = 2 cm? Show your work.

2

A label is a rectangle 25.12 cm by 9 cm. It wraps exactly once around a can. What are the radius and height of the can?

3

A round cake has a diameter of 20 cm and a height of 8 cm. You ice the top and the side, but not the bottom. What area do you ice?

Check your understanding

Quiz

1.

What shapes make up the net of a closed cylinder?

2.

In a cylinder's net, the length of the rectangle equals ...

3.

What is the surface area of a closed cylinder with r = 1 cm and h = 1 cm?

4.

A can has r = 2 cm and h = 10 cm. What is the area of a label covering its curved side?

5.

An open-top cylinder has how many circles in its net?

6.

Which unit is used for surface area?

For parents and teachers

Big idea: E2.6 asks students to represent cylinders as nets and determine their surface area by adding the areas of their parts. It builds on Grade 6 surface area of prisms and pyramids, and on this unit's circle lessons.

How to help at home: Peel the label off a sealed can and measure it. Cut open a cardboard tube. Ask, "Which part of the net is the label?" Real objects make the rectangle-circumference link click.

Common difficulties:

  • Not knowing the rectangle's length. It's the circumference, not the diameter.
  • Using the diameter in the circle area.
  • Counting the wrong number of circles for open containers and tubes.

Safety: never open a can or touch cut can edges; peel the paper label from a sealed can. Scissors with a grown-up.