At a glance. One lesson of about 60 minutes. It assumes the volume of a prism and cylinder and the slant height from lesson 1. Students use the formulas, explain the third, use the sphere in its cylinder, and find volumes of joined and recast solids.
Calculator: GDC optional. A GDC can hold a long product so that rounding happens once, but every calculation here can be written exactly as a multiple of π and done by hand.
What students will learn
Know
- Pyramid and cone: ⅓ × base area × vertical height.
- Sphere: 4⁄3πr³.
- A snug sphere is two thirds of its cylinder.
Understand
- A cube is six pyramids from its centre, so each is a sixth of the cube, which is a third of the half-cube box with the same base and height.
- Equal volumes give equations: melted solids keep their volume.
- A composite volume is a plan: parts, formulas, add or subtract.
Do
- Find the volume of a pyramid, cone, sphere and hemisphere.
- Plan and calculate the volume of a joined solid.
- Use equal volumes to find a missing length.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: volumes of solids (questions 2 to 8).
- Criterion B. Investigating patterns: the what-if, comparing surface areas and volumes in the snug cylinder.
- Criterion C. Communicating: the labelled plan for a composite solid, and the exit ticket that chooses a calculation.
- Criterion D. Applying mathematics in real-life contexts: the grain bin, the steel ball and the recast metal.
The one thing to do with the figure
Make the comparison before naming the ratio. Set r = 3 and ask students to divide each volume by the cone's volume. Only then ask what the figures say about the three solids.
Then ask: does this depend on r? Move the slider and watch the same three numbers reappear.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the tin. Collect predictions for the fractions. |
| 12 min | The figure; the key idea and the cube argument; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to the right-angled triangle for the height. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 90π cm³; 225 cm³; 5. |
|---|---|
| Hook | The ball fills two thirds of the tin and the cone one third. |
| Worked examples | 100π ≈ 314.2 cm³; about 3.6 litres; 84π ≈ 263.9 cm³. |
| Faded example | 56π ÷ 3 ≈ 58.6 cm³. |
| 1. Which height goes in the pyramid volume | C. A uses the slant height, B leaves out the third, D uses the whole base in the triangle. |
| 2. Volume of a pyramid | 336. ⅓ × 144 × 7. |
| 3. Volume of a sphere from its diameter | 904.8. 288π. |
| 4. Cylinder with a hemisphere | 1204.3. 300π + 250π ÷ 3. |
| 5. Melted and recast | 3. r³ = 27. |
| 6. The sphere in its cylinder | 360. ⅔ of 540. |
| 7. Equal volumes: cone and hemisphere | 2. h = 2r. |
| 8. A sphere in a cube | 52.4. π ÷ 6. |
| 9. Planning a composite volume | D. A uses a whole sphere, B a cylinder, C subtracts. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Used the slant height, left out the third, or used the whole base side in the triangle. | Sketch the triangle: vertical height, half the base, slant height. |
| 1008 or 28 in question 2 | Forgot the third, or did not square the base side. | Write the base area as a number first. |
| 7238.2, 452.4 or 150.8 in question 3 | Used the diameter as the radius, found the surface area, or squared instead of cubed. | Write r = 6 and the formula with r cubed. |
| 1466.1 or 942.5 in question 4 | Used a whole sphere, or left out the roof. | Name the parts before calculating. |
| 27 or 4.3 in question 5 | Did not take the cube root, or left out the third. | Write both volumes in terms of π and equate. |
| 180 or 270 in question 6 | Used a third, or guessed a half from the picture. | Return to the three volumes in the figure. |
| 4 or 0.7 in question 7 | Used a whole sphere, or left out the third. | Write the hemisphere as ⅔πr³ and cancel. |
| 78.5 or 26.2 in question 8 | Used the fraction for a circle in a square, or a hemisphere. | Side of the cube = 2r. |
| A, B or C in question 9 | Used a whole sphere, a cylinder, or subtracted. | Name each solid and say if it is added or taken away. |
Questions worth asking
- Why is a pyramid a third of its prism? A cube is six pyramids from the centre, each a sixth, which is a third of the box with the same base and height.
- Does the ratio of cone to sphere to cylinder depend on r? No: all three volumes contain r³, so the ratio 1 : 2 : 3 is fixed.
- How does a hemisphere's volume compare with a cone of equal radius and height 2r? They are equal, ⅔πr³.
- What if the sphere does not touch the lid? Then the cylinder is taller, and the ratio changes.
Differentiation
A way in for anyone stuck
- Use three containers of equal base and height: pour water from a cone into a cylinder three times.
- Give a table with columns solid, formula, measurements, volume.
- Draw each composite solid separated into its parts before any numbers are used.
Stretch
- A cone, a hemisphere and a cylinder have the same radius r and the same height r. Show that their volumes are in the ratio 1 : 2 : 3.
- A cone's vertical height is twice its radius. Show that its volume is half that of the sphere of the same radius.
Students learning in English as an additional language
“Volume” is how much space a solid fills. A “hemisphere” is half a sphere. “Recast” means melted and made into a new shape. “Capacity” is how much a container holds. Say “a third of” each time you use the pointed-solid rule.
| Word | Plain meaning | Example |
|---|---|---|
| volume | the amount of space a solid fills, in cubic units | a box 3 by 4 by 5 has volume 60 |
| hemisphere | half of a sphere | a bowl of radius 2.5 cm |
| capacity | the volume of liquid a container holds | 1 cm³ = 1 mL |
| recast | melted down and made into a new shape with the same volume | a candle melted into a ball |
| composite solid | a solid made by joining simpler solids | an ice-cream cone with a scoop |
Sentence starters for explaining aloud or in writing:
- The solid is made of … and …, so its volume is …
- The vertical height is found from …
- Both volumes are equal, so …
Two things not to say
“Volume is just area times height.” That is for prisms; a pointed solid is a third of it.
“A hemisphere is half, so halve the final answer.” Halve the formula, then check the radius.
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.
- PyramidA square pyramid has base side 6 cm and height 10 cm. Find its volume.
120 cm³. - SphereFind the volume of a sphere of radius 3 cm, in terms of π.
36π cm³.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA hemispherical dome in Isfahan has radius 9 m. Find the volume of air under it in terms of π.
486π m³. - Earlier work (Year 9)A cylinder has radius 4 cm and volume 80π cm³. Find its height.
5 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA sphere of radius 3 cm is melted and recast as a cone of the same radius. Find its height.
12 cm. - ProofShow that a cone with radius equal to its height has half the volume of the hemisphere of the same radius.
⅓πr³ is half of ⅔πr³.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
A calculator is optional. Keep answers exact as multiples of π until the final step.