At a glance. One lesson of about 60 minutes. It follows lesson 1 and assumes the area and circumference of a circle. Students use the net of a cylinder, find its surface area in terms of π and to one decimal place, handle open tins and pipes, and work back to a height.
Calculator: scientific calculator. The π key is needed for the one-decimal-place answers. Students add in terms of π first and press the key once, so rounding does not build up.
What students will learn
Know
- A cylinder's net is two circles and a rectangle.
- The rectangle is 2πr wide and h tall.
- The surface area of a closed cylinder is 2πr² + 2πrh.
Understand
- The curved side looks like a different kind of thing, but it flattens to a plain rectangle.
- A formula is a list of faces: remove a face and remove its term.
- A short, wide cylinder is mostly ends; a tall, thin one is mostly side.
Do
- Find the area of a closed or open cylinder in terms of π and to one decimal place.
- Find the area of a label or a pipe, which is the curved side only.
- Work back from a curved surface area to a height.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the net and the surface area formulas (questions 1 to 9).
- Criterion B. Investigating patterns: the second figure. Ask students to predict the total before each toggle and then explain the missing face.
- Criterion C. Communicating: sketching the net with its measurements, as in the worked example, and naming which faces are present.
- Criterion D. Applying mathematics in real-life contexts: tomato tins, a bucket, a pipe and a paint tin (the hook, the worked example, questions 5, 7 and 8).
The one thing to do with the figure
Unroll before you calculate. Ask students to say how wide the rectangle will be before they press Unroll, then read the width from the panel. The width equals the distance round the circle, which is the circumference from Year 8.
In the second figure, ask “What has gone?” each time a button is pressed. The answer is a face, and the face has a number: take it off the total.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: the label that was too short. Collect predictions of the width. |
| 10 min | First figure: unroll, then change the radius and height and read the three areas. |
| 10 min | Key idea; second figure; the closed tin worked example; students finish the pipe in pairs. |
| 5 min | Question 1, the check. If many choose B or D, return to the circumference in the net. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 6π cm; 52 cm²; 42. |
|---|---|
| Hook | The label must be 6π ≈ 18.8 cm wide; 6 cm is only the width of the tin. |
| Worked example | 170π ≈ 534.1 cm². |
| Faded example | 150π ≈ 471.2 cm². |
| 1. A side of the net | C. A is the radius, B the diameter, D half the circumference (πr). |
| 2. Area of the curved side | 56. 2π × 4 × 7 = 56π. |
| 3. Area of the two ends | 32. 2 × 16π = 32π. |
| 4. Whole surface to one decimal place | 565.5. 180π = 565.48… |
| 5. An open tin | 1125. 225π + 900π. |
| 6. Working back to the height | 15. 150π ÷ 10π = 15. |
| 7. A label round a tin | 276.5. 88π = 276.46… |
| 8. Open tin from a diameter | 1885.0. 600π = 1884.96…, which is 1885.0 to one decimal place. |
| 9. Ends against the curved side | B. Ends 200π, curved side 60π. A trusts how the shape looks; C uses the diameter as the radius and counts one end; D has the right conclusion with the 2 missed twice. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Used the radius, the diameter or half the circumference for the side of the rectangle. | Wrap a strip of paper round a tin and measure it against the diameter: three diameters and a bit. |
| 28, 112 or 15 in question 2 | Missed the 2, squared the radius, or added. | Draw the rectangle with its two sides marked, then multiply them. |
| 16, 64 or 8 in question 3 | Used one circle, put the 2 inside the square, or did not square. | Write the area of one circle as its own line before doubling it. |
| 339.3, 452.4 or 226.2 in question 4 | Left out the ends, counted one end, or left out the side. | Ask for the list of faces first and tick each one as it is counted. |
| 1350, 900 or 675 in question 5 | Counted a lid that is not there, left out the base, or missed the 2. | Ask what the bucket looks like from above and from below: is there a face on each? |
| 30, 75 or 6 in question 6 | Divided by the radius only, by 2 only, or by r². | Write the area as (2πr) × h and ask what the width is. |
| 138.2, 100.5 or 377.0 in question 7 | Missed the 2, found the ends, or the whole surface. | Ask: what does the label cover and what does it not? |
| 4398.2, 2199.1 or 1570.8 in question 8 | Used the diameter as the radius, counted the lid, or left out the base. | Cross out 20 and write the radius before anything else; then list the faces. |
| A, C or D in question 9 | Trusted the look, used the diameter as the radius and counted one end, or dropped a 2. | Work out the area of a single circle and of the rectangle, side by side. |
Questions worth asking
- Why is the width of the rectangle the circumference? Listen for “the side wraps once round the circle, so its edge is the same length as the circle's edge”.
- What does the formula 2πr(r + h) tell you? It is the same area with 2πr taken out as a factor; r + h is the radius of the circle plus the height.
- A cylinder is tall and thin. Which part of its surface is bigger? The curved side: the rectangle is wide enough and tall; the circles are small.
- Which faces does a drinking straw have? Only the curved side; it is a pipe, open at both ends.
Differentiation
A way in for anyone stuck
- Give students a card net to cut and roll so that the rectangle's width visibly matches the circle's edge.
- Provide a list of faces to tick for each question: lid, base, curved side.
- Use questions 2 and 3 first to give the two parts on their own before any total.
Stretch
- Question 9, then: for what height is the area of the curved side equal to the area of the two ends of a cylinder with radius r? (h = r.)
- Write the surface as 2πr(r + h). A closed cylinder has r = 3 cm and surface area 60π cm². Find h. (h = 7: 6π(3 + h) = 60π.)
Students learning in English as an additional language
“Net” means a flat shape that folds into a solid. “Curved” side and “flat” ends are two different words to learn. Have students say “two circles and a rectangle” each time they find the surface area of a cylinder.
| Word | Plain meaning | Example |
|---|---|---|
| net | a flat shape that folds up into a solid | two circles and a rectangle make a cylinder |
| surface area | the total area of all the faces of a solid | 170π cm² |
| circumference | the distance round a circle, 2πr | 6π cm |
| curved side | the part of a cylinder between the two ends | the label on a tin |
| open tin | a cylinder with no lid | a bucket |
| pipe | a cylinder with both ends open, so only the curved side | a drinking straw |
Sentence starters for explaining aloud or in writing:
- The net has … circles and a rectangle that is … by …
- The width of the rectangle is … because …
- This shape has no …, so I take away …
Two things not to say
“Use πr for the width.” That is half the way round; the rectangle is 2πr wide.
“Surface area is the formula 2πr² + 2πrh.” It is the faces that are present; the formula is only the list for a closed tin.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- CirclesA cylinder has radius 3 cm. Find the area of its two circular ends in terms of π.
18π cm². - SideA cylinder has radius 2 cm and height 10 cm. Find its curved side in terms of π.
40π cm².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA tin of beans has radius 3.5 cm and height 10 cm. Find the area of its label to one decimal place.
2π × 3.5 × 10 = 70π = 219.9 cm². - Earlier work (Year 8)The surface area of a cuboid 3 cm by 4 cm by 10 cm?
2 × (12 + 30 + 40) = 164 cm².
3Challenge
Reasoning, generalizing and working backwards.
- Working backA closed cylinder has radius 4 cm and surface area 112π cm². Find its height.
32π + 8πh = 112π, so h = 10 cm. - ReasoningThe radius of a closed cylinder doubles and its height stays the same. Is its surface area doubled?
No: the ends become 4 times as big, the side only 2 times. The total is between 2 and 4 times.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or button, then the arrow keys or the space bar) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use a tin with the label peeled off: it lies flat as a rectangle, and the lid and the base can be traced.