Year 9 · Unit 4 Cylinders, cones and spheres · Lesson 2 of 6

What does a cylinder look like when you unroll it?

Peel the label off a tin and it lies flat as a rectangle. Its width is the distance round the circle, so the whole surface is two circles and one rectangle: 2πr² + 2πrh.

Geometry and TrigonometryScientific calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 8 (circles): a circle has radius 3 cm. What is its circumference, in terms of π?
    Answer6π cm. The circumference is 2 × π × 3.
  2. From Year 7 (nets and surface area): a cuboid is 2 cm by 3 cm by 4 cm. What is its surface area?
    Answer52 cm². The faces are 2 × (6 + 8 + 12) = 2 × 26 = 52.
  3. Work out 2 × 3² + 2 × 3 × 4.
    Answer42. Square first: 2 × 9 = 18, then 2 × 3 × 4 = 24, and 18 + 24 = 42.
Words used in this lesson
net
a flat shape that folds up into a solid. For example: two circles and a rectangle make a cylinder.
surface area
the total area of all the faces of a solid. For example: 170π cm².
circumference
the distance round a circle, 2πr. For example: 6π cm.
curved side
the part of a cylinder between the two ends. For example: the label on a tin.
open tin
a cylinder with no lid. For example: a bucket.
pipe
a cylinder with both ends open, so only the curved side. For example: a drinking straw.

The label that was too short

A food company in Lima prints labels for tins of tomatoes. Each tin has radius 3 cm and height 10 cm. A designer cuts a label 6 cm wide and 10 cm tall, because the tin is 6 cm across.

Predict first: will the label go all the way round the tin? If not, how wide must it be? Then unroll a cylinder in the figure and watch what the curved side turns into.

2 cm
7 cm
4π ≈ 12.6 cmwidth of the rectangle (2πr)
28π ≈ 88.0 cm²area of the rectangle (curved side)
36π ≈ 113.1 cm²area of the whole surface

Radius 2 cm, height 7 cm: the two circles are 2 × π × 2² = 8π cm², the curved side is 2π × 2 × 7 = 28π cm², and the whole surface is 36π ≈ 113.1 cm².

Show me the steps
  1. Radius 2 cm, height 7 cm. Two circles: 2 × π × 2² = 8π cm². Curved side: 2π × 2 × 7 = 28π cm². Whole surface: 36π ≈ 113.1 cm².
  2. Press Unroll into a net. The curved side opens into a rectangle 7 cm tall and 2π × 2 = 4π ≈ 12.6 cm wide, with the two circles above and below it. Its area is 4π × 7 = 28π cm², as before.
  3. Make the radius 5 cm and keep the height 7 cm: two circles 2 × π × 5² = 50π, rectangle 2π × 5 × 7 = 70π, whole surface 120π ≈ 377.0 cm².
  4. Make the radius 1 cm and the height 10 cm: two circles 2π, rectangle 2π × 1 × 10 = 20π, whole surface 22π ≈ 69.1 cm². A thin, tall cylinder is almost all rectangle.

A cylinder's net is two circles and one rectangle. The rectangle is as tall as the cylinder and as wide as the distance round the circle, which is the circumference 2πr.

So the area of the curved side is 2πr × h, and the whole surface is 2πr² + 2πrh: the two circles plus the rectangle.

For an open tin or a pipe, take away the circles that are not there. The formula is a list of faces.

66π ≈ 207.3 cm²total surface area
closed cylinderwhat it is

Lid on and base on: a closed cylinder. 9π + 9π + 48π = 66π ≈ 207.3 cm².

Show me the steps
  1. Lid on and base on, a closed cylinder: 9π + 9π + 48π = 66π ≈ 207.3 cm².
  2. Take the lid off, an open tin: 9π + 48π = 57π ≈ 179.1 cm². One circle is gone, and the curved side stays.
  3. Take the base off as well, a pipe: 48π ≈ 150.8 cm². Only the curved side is left, a rectangle.
  4. Put the lid back and take the base off: 9π + 48π = 57π ≈ 179.1 cm² again. It does not matter which end is missing.

A common mistake: using πr (half the circumference) for the width of the rectangle, or forgetting the circles at the ends. Another is adding only one circle: a closed cylinder has a lid and a base, so there are two.

Worked example: a closed tin

A closed cylinder has radius 5 cm and height 12 cm. Find its whole surface area in terms of π, and to one decimal place.

  1. The two circles: 2 × π × 5² = 50π cm².
  2. The curved side is a rectangle 12 cm tall and 2π × 5 = 10π cm wide: 10π × 12 = 120π cm².
  3. Add: 50π + 120π = 170π cm².
  4. With the π key: 170 × π = 534.07…, so the surface area is 534.1 cm² to one decimal place.

Your go. The first step is given. A metal pipe is open at both ends. It has radius 3 cm and length 25 cm. Find the area of its outside surface, in terms of π and to one decimal place.

  1. The pipe has no circular ends, so only the curved side counts. The rectangle is as wide as the circumference: 2π × 3 = 6π cm.
  2. Now multiply by the length, then use the π key to check.
Check the answer

The area is 6π × 25 = 150π cm². The calculator gives 471.23…, so 471.2 cm² to one decimal place.

With a calculator

Write the exact answer first, then use the π key at the end. For a closed cylinder with radius 5 cm and height 12 cm, the surface area is 2π × 5² + 2π × 5 × 12 = 50π + 120π = 170π cm². Press 170 × π: 534.07…, so 534.1 cm² to one decimal place.

Add the two parts in terms of π before you press anything. Then you press the π key once, and rounding does not build up. A quick check: 3 × 170 = 510, so an answer near 534 is sensible.

Check before you go on

1. A cylinder has radius 4 cm and height 9 cm. In its net, one side of the rectangle is 9 cm. What is the length of the other side?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A cylinder has radius 4 cm and height 7 cm. Its curved side has area ?π cm². What number goes in front of π?

3. The same cylinder has radius 4 cm and height 7 cm. The two circular ends together have area ?π cm². What number goes in front of π?

Level 2 · Building

4. A closed cylinder has radius 6 cm and height 9 cm. Find its whole surface area in cm², to one decimal place.

5. A metal bucket is a cylinder with no lid, with radius 15 cm and height 30 cm. The metal in it has area ?π cm². What number goes in front of π?

6. A cylinder has radius 5 cm and a curved side of area 150π cm². What is its height, in cm?

Level 3 · Stretch

7. A label wraps once round the curved side of a tin, with no overlap. The tin has radius 4 cm and height 11 cm. How much paper is there in the label, in cm², to one decimal place?

8. A paint tin has no lid. It is 20 cm across and 25 cm tall. How much sheet metal makes it, in cm², to one decimal place?

Something to think about

A closed cylinder has a height equal to its diameter. How many times as big as its base area is its whole surface area?

One way to think about it

Let the radius be r, so the height is 2r. The two ends are 2πr² and the curved side is 2πr × 2r = 4πr², so the whole surface is 6πr². The base area is πr², so the surface is 6 times the base area. This is true whatever r is: the shape is the same, only its size changes.

Exit ticket

9. A closed cylinder is short and wide: radius 10 cm and height 3 cm. Which statement is correct, including its numbers?