At a glance. One lesson of about 60 minutes. It assumes the area of a circle and the volume of a prism from Years 7 and 8. Students find the volume of a cylinder in terms of π and to one decimal place, work back to a missing height, and explain why doubling the radius gives four times the volume.
Calculator: scientific calculator. The π key is needed for the one-decimal-place answers. Most of the lesson keeps π as a symbol, so the thinking is about structure and not about buttons.
What students will learn
Know
- The volume of a prism is the area of its base times its height.
- The volume of a cylinder is V = πr²h.
- An answer in terms of π keeps π as a symbol, such as 160π cm³.
Understand
- A cylinder is a prism whose base is a circle, so no new idea is needed, only a new base area.
- The radius is squared, so a change to the radius has a much bigger effect than the same change to the height.
- Working in terms of π is exact; a decimal is an approximation.
Do
- Find the volume of a cylinder from its radius or diameter and height, in terms of π and to one decimal place.
- Find a missing height from a volume and a radius.
- Predict the effect of doubling or halving a measurement on the volume.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the volume formula and its use, in terms of π and to one decimal place (questions 1 to 9).
- Criterion B. Investigating patterns: the second figure. Ask students to predict the volume factor for each change before they move the slider, then say why the radius and the height behave differently.
- Criterion C. Communicating: writing the radius, then the base area, then the volume in terms of π, then the rounded answer, as in the worked example and question 9.
- Criterion D. Applying mathematics in real-life contexts: cooking pots, tins, a water bottle, a tank and a candle in a box (the hook, the worked example, questions 4, 7 and 8).
The one thing to do with the figure
Make students predict before they move. In the second figure, ask “What will the volume be if the radius doubles? Write a number, then check it.” Most write 40π. Settle it with the figure, then ask why the height and radius are different.
In the first figure, push the radius to 6 and the height to 10 and ask for the number of layers and the area of each: the volume is a product of two ideas, and students can read both off the panel.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: Pot A and Pot B. Collect predictions and reasons; do not settle them yet. |
| 10 min | First figure: stack the layers. Ask for the area of one layer and the number of layers before reading the panel. |
| 10 min | Key idea; second figure with predictions; the chickpea tin worked example; students finish the bamboo pipe in pairs. |
| 5 min | Question 1, the check. If many choose A or C, return to the radius and diameter warning. |
| 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, then settle the hook. |
The answers
| Warm-up | 25π cm²; 120 cm³; 160. |
|---|---|
| Hook | Pot A holds 360π cm³ and Pot B holds 180π cm³: Pot B holds half as much. |
| Worked example | 160π ≈ 502.7 cm³. |
| Faded example | 1000π ≈ 3141.6 cm³. |
| 1. Which expression is the volume? | B. A forgets to square the radius; C uses the diameter 10 as the radius; D is the curved surface area, 2πrh. |
| 2. Base area times height | 600. Base area times height: 50 × 12. The end can be any shape. |
| 3. Volume in terms of π | 36. π × 2² × 9 = 36π. |
| 4. Volume to one decimal place | 1005.3. 320π = 1005.30… |
| 5. Diameter given | 864. The radius is 12: π × 144 × 6 = 864π. |
| 6. Working back to the height | 8. 200π ÷ 25π = 8 cm. |
| 7. Decimal radius | 28.3. 9π = 28.27… |
| 8. Cylinder inside a box | 214.6. 1000 − 250π = 1000 − 785.4. |
| 9. Which cylinder holds more? | C. A: 72π, B: 72π. A forgets to square; B ignores the height; D uses height alone. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or C in question 1 | Did not square the radius, or used the diameter as the radius. | Ask: what shape is the base, and what is the formula for its area? Which measurement does that formula use? |
| 62 or 300 in question 2 | Added the base area and height, or halved the triangle again. | Ask what the layers are: how many, and how big is each? The 50 is already the area of one layer. |
| 18, 144 or 13 in question 3 | Did not square, used the diameter, or added. | Write r², then × h, as two separate lines before any number goes in. |
| 251.3, 4021.2 or 502.7 in question 4 | Did not square the radius, used the diameter, or found the curved surface. | Ask for the radius and the base area as separate answers before the volume. |
| 3456, 72 or 150 in question 5 | Squared the diameter, did not square the radius, or added the squared radius and the height. | Cross out 24 and write 12 beside it before anything else. |
| 40, 5000 or 175 in question 6 | Did not square the radius when dividing, multiplied, or subtracted the base area. | Draw the layers: volume 200π is 25π per layer, so how many layers? |
| 18.8, 113.1 or 9 in question 7 | Did not square the radius, doubled it, or left out π. | Ask what 1.5 squared is. Decimals hide the squaring slip. |
| 785.4, 1000 or 2141.6 in question 8 | Gave only one of the two volumes, or used the diameter as the radius so the candle came out bigger than the box. | Ask what the question is asking for: the space that is not the candle. |
| A, B or D in question 9 | Multiplied radius by height, ignored the height, or used height alone. | Ask for the base area of each cylinder first, then multiply by the height. |
Questions worth asking
- Why is the radius squared and the height not? Listen for “the base is an area, so two lengths multiply, and then the height is one more length”.
- Which is more powerful, doubling the radius or doubling the height? The radius: it gives four times the volume against two times. The volume is 3 lengths multiplied, but two of them are the radius.
- When would you leave an answer in terms of π, and when would you round? Terms of π when you may need to go on calculating or want an exact value; round when the answer is for a measurement a person will use.
- Is a cylinder really a prism? Yes: a prism has the same cross-section all along its length. The circle is the cross-section here.
Differentiation
A way in for anyone stuck
- Give a formula frame: V = π × (…)² × (…), with boxes for the radius and the height.
- Ask students to underline the word radius or diameter in each question and write the radius before anything else.
- Use whole-number radii only at first; questions 2 and 3 and the worked examples need no calculator.
Stretch
- Question 9, then: cylinder C has radius 6 cm and height 8 cm. Compare it with A and B without calculating the volume of C.
- A cylinder's height equals its diameter and its volume is 54π cm³. Find its radius.
Students learning in English as an additional language
“Radius” and “diameter” are often swapped by mistake, so say them together: the radius is half the diameter. “In terms of π” means leave the symbol π in the answer. Have students say “area of the base times the height” each time they use V = πr²h.
| Word | Plain meaning | Example |
|---|---|---|
| cylinder | a solid with two equal circles as its ends and a curved side | a tin of beans |
| radius | the distance from the centre of a circle to its edge | 5 cm |
| diameter | the distance across a circle through its centre, twice the radius | 10 cm |
| base area | the area of the face the solid stands on | π × 5² = 25π cm² |
| height | how tall the solid is, at right angles to its base | 8 cm |
| volume | how much space a solid takes up | 200π cm³ |
| in terms of π | written with the symbol π instead of a decimal | 160π cm³ |
Sentence starters for explaining aloud or in writing:
- The radius is …, so the base area is …
- The volume is the base area times …, which is …
- When the radius doubles, the volume is multiplied by … because …
Two things not to say
“Use 3.14 for π.” The π key is more accurate and the answers on the page assume it; 3.14 changes the last digit.
“Double the size and the volume doubles.” It does only when one length is doubled; the radius gives four times.
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.
- Base areaA cylinder has base area 30 cm² and height 7 cm. Find its volume.
210 cm³. - Terms of πA cylinder has radius 5 cm and height 2 cm. Find its volume in terms of π.
50π cm³.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cylindrical water tank has diameter 2 m and height 3 m. Find its volume in m³, to one decimal place.
3π = 9.4 m³. - Earlier work (Year 8)A circle has radius 7 cm. Find its circumference in terms of π.
14π cm.
3Challenge
Reasoning, generalizing and working backwards.
- Working backA cylinder has volume 108π cm³ and height 12 cm. Find its radius.
r² = 9, so r = 3 cm. - ReasoningA cylinder's radius is tripled and its height is divided by 3. What happens to its volume?
Multiplied by 9, then divided by 3: it is multiplied by 3.
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, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Stack coins or cut a circle from card and trace it, layer by layer, to show the volume as base times height.