At a glance. One lesson of about 60 minutes. It follows lessons 1 to 5 and uses the cylinder, cone and hemisphere formulas together. Students split composite solids, handle litres and cm³, leave out joined faces, and compare designs of equal volume by their surface area.
Calculator: scientific calculator. The π key is needed for the one-decimal-place answers and for the equal-volume design comparison. Students add the parts in terms of π first and press the key once, so no rounding builds up.
What students will learn
Know
- The volume of a composite solid is the sum of the volumes of its parts.
- The surface area counts only the faces on the outside: joined circles are left out.
- 1 litre is 1000 cm³, and equal volumes are compared by surface area.
Understand
- Capacity and surface area are different measurements of the same solid, and a designer weighs them against each other.
- For a given volume, the best cylinder is neither the tallest nor the widest, but about as tall as it is wide.
- A joined face disappears from the surface but the volume carries on.
Do
- Find the volume and the outside surface area of a solid made of two or three standard solids.
- Convert between cm³ and litres.
- Choose between designs with the same volume by comparing their surface areas.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the formulas used together, and the unit conversion (questions 1 to 9).
- Criterion B. Investigating patterns: the first figure. Ask students to predict whether the metal needed will fall, rise, or stay about the same as the radius grows, then explain the minimum.
- Criterion C. Communicating: listing the parts and the faces before calculating, as in the worked example and question 9.
- Criterion D. Applying mathematics in real-life contexts: a tin of olive oil, a grain bin, a rocket toy, an ice-cream cone, a capsule, drinks cans and a silo (the hook, the worked and faded examples, and questions 2 to 8).
The one thing to do with the figure
Ask for the prediction, then the answer in words. Before moving the slider: “Does the metal needed fall as the tin gets wider, or rise?” Most say one or the other. The figure shows both, and the shallow bottom. Ask why a wide tin needs more: its circles are large.
In the second figure, ask “which circles are on the outside?” for each solid. The answer is none of the joined ones. Write the surface list before reading the rows.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: the one-litre tin. Collect predictions of the tall or the wide tin. |
| 10 min | First figure: seven radii and the shallow minimum. Return to the hook. |
| 10 min | Key idea; second figure and the surface lists; the grain bin worked example; students finish the rocket in pairs. |
| 5 min | Question 1, the check. If many choose C or D, return to the joined faces. |
| 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 | 18π cm³; 15π cm²; 2.5 litres. |
|---|---|
| Hook | Neither extreme: the radius 5 cm tin (12.7 cm tall) needs the least of the seven, 557.1 cm². |
| Worked example | Volume 52π/3 ≈ 54.5 m³; outside surface 20π ≈ 62.8 m². |
| Faded example | 102π ≈ 320.4 cm³. |
| 1. Which faces count? | B. A is a closed cylinder; C counts the joined circles; D counts the joined circles too. |
| 2. Volume of a cylinder and a hemisphere | 504. 360π + 144π. |
| 3. Litres from cubic centimetres | 942.5. 300 000π = 942 477.8 cm³. |
| 4. Cone with a hemisphere | 240. 96π + 144π. |
| 5. Outside surface of the cone and hemisphere | 414.7. 132π = 414.69… |
| 6. Surface of a capsule | 175.9. 56π = 175.93… |
| 7. Which can uses less metal | 12.6. 90π − 86π = 4π = 12.57… |
| 8. A silo with a cone roof | 552.9. 176π = 552.92… |
| 9. Choosing a design | A. P: 96π, Q: 160π. B compares the curved sides only; C says same volume means same card; D counts one end each. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Counted the joined circles, or counted whole solids. | Hold two real solids against each other: which faces can you still touch? |
| 360, 648 or 144 in question 2 | Left out the hemisphere, used a whole sphere, or left out the cylinder. | Name the parts before calculating: cylinder, and what else? |
| 942477.8, 94.2 or 0.9 in question 3 | Did not convert, used the cm² to m² factor, or changed to m³ instead of litres. | Ask: how many cm³ in one litre? Write that beside the answer. |
| 384, 432 or 264 in question 4 | Used a whole sphere, forgot the third, or used the slant height as the height. | Ask which length goes in a volume: the one inside the cone. |
| 527.8, 377.0 or 188.5 in question 5 | Counted the joined circle, used the height in the slant formula, or left out the dome. | Write a list of outside faces and tick each. |
| 201.1, 125.7 or 50.3 in question 6 | Counted the flat circles, or gave one part only. | Ask: does the capsule have a flat face you could touch? |
| 56.5, 270.2 or 282.7 in question 7 | Compared only the curved sides, or gave one area instead of the difference. | Ask what the question is asking for: the difference between the two. |
| 502.7, 653.5 or 586.4 in question 8 | Left out the roof, missed the third, or used the slant height. | Draw the silo and mark the heights: which is the roof's own height? |
| B, C or D in question 9 | Compared the curved sides only, thought the same volume means the same card, or counted one end each. | Both cartons hold the same: list the faces of each and add them all. |
Questions worth asking
- Why is the metal needed not smallest for the tallest or the widest tin? Tall: the curved side is big. Wide: the circles are big. The middle balances them.
- When two solids are glued together, what happens to the surface area? It is less than the sum, because the joined faces are hidden.
- Which uses less material for the same volume, a sphere or a cube? The sphere. That is why bubbles are round; a cube needs more surface.
- Why might a company not choose the best design? Handling, stacking, shelf space, cost of printing, and what customers expect.
Differentiation
A way in for anyone stuck
- Give a table to fill: part, shape, formula, value, then total volume and a list of outside faces.
- Provide the formulas on a card, including 1 litre = 1000 cm³.
- Use only whole-number answers in terms of π for the first two composites.
Stretch
- Question 9, then: a closed carton holds 128π cm³ and its height equals its diameter. Find its radius and its card. (2πr³ = 128π, so r = 4 cm and h = 8 cm: this is carton P, 96π cm². The carton whose height equals its diameter is the better one.)
- A sphere of radius 3 cm is dropped into a cylinder of radius 3 cm and height 6 cm full of water. How much water spills, and what fraction of the cylinder is left? (36π cm³; one third.)
Students learning in English as an additional language
“Composite” means made of more than one part. “Capacity” is how much a container holds, the same as volume, usually in litres. “Outside surface” means the faces you can touch. Have students say the parts and the faces aloud before they calculate.
| Word | Plain meaning | Example |
|---|---|---|
| composite solid | a solid made of two or more simple solids joined together | a capsule |
| capacity | how much a container holds | 1 litre |
| litre | a unit of capacity, 1000 cm³ | a bottle of water |
| hemisphere | half of a sphere | a bowl |
| surface area | the total area of the outside faces | 84π cm² |
| design | a choice of shape and size to do a job | a one-litre tin |
Sentence starters for explaining aloud or in writing:
- The solid is made of … and …, so its volume is …
- The faces on the outside are …; the … is inside, so I leave it out.
- I would choose design … because it uses less … for the same …
Two things not to say
“Add the surface areas of the parts.” Joined faces are hidden, so the total is less.
“The biggest surface holds the most.” Capacity is volume. Two shapes with the same volume can have different surface areas.
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.
- ConvertA bottle holds 750 cm³. How many litres is that?
0.75 litres. - PartsA cylinder has volume 20π cm³ and a cone on top has volume 4π cm³. Find the total in terms of π.
24π cm³.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA water tower is a cylinder of radius 3 m and height 5 m with a hemisphere of radius 3 m on top. Find its volume in terms of π.
45π + 18π = 63π m³. - Earlier work (Year 9)A sphere has radius 6 cm. Find its surface area in terms of π.
144π cm².
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA closed cylinder has radius r and height 2r. Show that its surface area is 6πr² and its volume is 2πr³.
Ends 2πr², side 2πr × 2r = 4πr², total 6πr²; volume πr² × 2r = 2πr³. - Working backA cylinder of radius 3 cm has a hemisphere of radius 3 cm on top. The total volume is 72π cm³. Find the cylinder's height.
Hemisphere 18π, so the cylinder is 54π = 9π × h, h = 6 cm.
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? Bring in a few real tins and a tape measure. Measure each, work out the volume, and compare with the label.