At a glance. One lesson of about 60 minutes. It assumes the volume of prisms and cylinders and the work on slant height from lesson 1. Students find the volume of pyramids, cones, spheres, hemispheres and simple composite solids, in exact form and rounded.
Calculator: GDC optional. The lesson is done by hand with π left in exact answers. The optional section shows GDC users how to enter a sphere volume in one expression and round once, at the end.
What students will learn
Know
- Pyramid and cone: V = ⅓ × base area × vertical height.
- Sphere: V = 4πr³ ÷ 3. Hemisphere: 2πr³ ÷ 3.
- Volume uses the vertical height, not the slant height.
Understand
- A pyramid or cone is a fixed fraction of the prism or cylinder on the same base.
- A sphere fills two thirds of the cylinder that just holds it, so its formula is related to the cylinder's.
Do
- Find the volume of pyramids, cones and spheres in exact form and rounded.
- Work back from a volume to a height.
- Find the volume of a composite solid made from a cone and a hemisphere.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: volume formulae for pyramids, cones, spheres and hemispheres (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students find that the ratios do not change with size and describe why.
- Criterion C. Communicating: writing the formula, then the substitution, then the answer on three separate lines (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the kulfi cone, the water tank and the ice-cream cone.
The one thing to do with the figure
Settle the hook with the shaded band. Set a radius and a height, and ask students to read the cone and the cylinder volumes. Then ask: if the shaded layer has the cone's volume, what fraction of the height is it?
In the second figure, ask students to change r and notice the fraction does not change. Ask why: both volumes contain r³.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: cones and cup. Collect votes and reasons. |
| 12 min | First figure; key idea; settle the hook; the sphere figure. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask what a cone of the same height holds, then a cone three times as tall. |
| 20 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 | 45π cm³; 64; 41 cm. |
|---|---|
| Hook | 3 cones fill the cup, because the cone is one third of the cylinder. |
| Worked examples | 384 cm³; 628.3 cm³; 2304π ≈ 7238.2 cm³. |
| Faded example | 120 cm³. |
| 1. A cone and its cylinder | C. A ignores the height; B leaves out the third; D divides for the height. |
| 2. Volume of a pyramid | 270. ⅓ × 81 × 10. |
| 3. Volume of a cone | 84. ⅓ × 36 × 7. |
| 4. Volume of a sphere in terms of π | 288. 4 × 216 ÷ 3. |
| 5. A sphere from its diameter | 1437 m³. 4π × 343 ÷ 3. |
| 6. Working back to a height | 12. 3h = 36. |
| 7. A hemisphere bowl | 1527 cm³. 486π. |
| 8. A cone topped with a hemisphere | 48. 30π + 18π. |
| 9. Volume of a cone | B. A has no third; C uses the slant height; D does not square r. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Ignored the height, left out the third, or divided for the height. | Write ⅓πr²h for the cone with h three times the cylinder's, side by side with πr²h. |
| 810 or 30 in question 2 | Forgot the one third, or forgot to square the base edge. | Write the base area on its own line first. |
| 252 or 14 in question 3 | Gave the cylinder, or forgot to square r. | Write r² = 36 before multiplying. |
| 864 or 144 in question 4 | Forgot to divide by 3, or used the surface area formula. | Compare 4πr³ ÷ 3 with 4πr²: one has a cube. |
| 11494 or 4310 in question 5 | Used the diameter as the radius, or forgot to divide by 3. | Write r = ____ on the first line of every sphere question. |
| 4 or 108 in question 6 | Forgot the third, or did not divide by r². | Write the equation ⅓ × 9 × h = 36 and solve it in two lines. |
| 3054 or 4580 in question 7 | Used the whole sphere, or forgot to divide by 3. | Find the sphere first, then halve it. |
| 30, 66 or 108 in question 8 | Left out the hemisphere, used a whole sphere, or forgot the cone's third. | Draw the two solids apart and find each volume. |
| A, C or D in question 9 | Left out the third, used the slant height, or did not square r. | Draw the right-angled triangle inside the cone and label h before using the formula. |
Questions worth asking
- Why is the cone one third and not one half of the cylinder? Listen for: a cube can be cut into three identical square-based pyramids, each with the cube's base and height. Accept the sand experiment.
- Why does the sphere need a cube and its surface area a square? Volume has three lengths multiplied, area has two.
- Does the slant height matter for volume? No: volume uses the vertical height, the slant is only needed for the surface.
- How could you find a cone's volume if you only knew its slant height and radius? Find h from Pythagoras first, then use the volume formula.
Differentiation
A way in for anyone stuck
- Provide a formula card with a labelled picture of each solid, marking r and the vertical height.
- Use physical solids (a cone and cylinder with the same base) and rice or sand.
- Ask students to write the substitution on its own line before using the calculator.
Stretch
- A cone and a hemisphere have the same radius r, and the cone's vertical height is r. Show that the hemisphere holds twice as much.
- A cone of radius 9 cm and slant height 15 cm: find its volume in terms of π.
Students learning in English as an additional language
“A third” and “one third” mean the same: 1 divided by 3. “Cube” here is a verb: to cube a number is to multiply it by itself three times. Say “four thirds pi r cubed” aloud with the class.
| Word | Plain meaning | Example |
|---|---|---|
| volume | the amount of space inside a solid | cm³ or litres |
| sphere | a perfectly round solid, like a ball | a football |
| hemisphere | half of a sphere | a bowl |
| pyramid | a solid with a flat base and triangular faces that meet at a point | a roof |
| composite solid | a solid made from two or more simpler solids | a silo: a cylinder with a dome |
| exact form | an answer left with π in it, not rounded | a volume of 20π cm³ |
Sentence starters for explaining aloud or in writing:
- The volume is one third of … because …
- The radius is … because the diameter is …
- I will find the volume of each part and then …
Two things not to say
“It is the same formula as the cylinder.” The cone has the extra third, and the sphere has a different formula.
“Use 3.14.” Keep π exact until the last step.
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 pyramid has a square base of edge 6 cm and vertical height 7 cm. Find its volume.
84 cm³. - Earlier work (lesson 1)A cone has radius 12 cm and vertical height 35 cm. Find its slant height.
37 cm.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA conical mound of grain in a market in Dakar has radius 3 m and vertical height 2 m. Find its volume to the nearest whole number.
6π ≈ 19 m³. - Working backA sphere has volume 36π cm³. Find its radius.
r³ = 27, so r = 3 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA cone and a cylinder have the same volume and the same radius. How do their heights compare?
The cone's height is 3 times the cylinder's. - OpenA cone and a hemisphere have the same volume. Find a radius and a height that make this true.
For example r = 3 for both: hemisphere 18π, cone ⅓ × 9 × h, so h = 6.
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. “Show me the steps” gives the same states in words.
No device? A cone and cylinder of the same base and some sand are enough. The With a GDC section is optional.