At a glance. One lesson of about 60 minutes. It follows the cone lesson. Students find the surface area and volume of a sphere and a hemisphere in terms of π, work back from a surface area to a volume, and see the cone, sphere and cylinder in the ratio 1 : 2 : 3. Cubes of whole numbers are the heaviest arithmetic.
Calculator: no calculator. No calculator is needed: every answer is in terms of π and the cubes are small whole numbers. A calculator would only tempt students to round too early.
What students will learn
Know
- The surface area of a sphere is 4πr² and its volume is 4/3 πr³.
- A solid hemisphere has curved surface 2πr², whole surface 3πr² and volume 2/3 πr³.
- A cone, a sphere and a cylinder with the same radius and a height of 2r have volumes in the ratio 1 : 2 : 3.
Understand
- The surface of a sphere covers exactly four great circles; this is a fact worth seeing, not just memorizing.
- The volume has r cubed and the area r squared, so halving a radius gives one eighth of the volume but one quarter of the area.
- The three solids are connected: the sphere is two thirds of the cylinder that fits round it.
Do
- Find the surface area and volume of a sphere or hemisphere, in terms of π.
- Work back from a surface area to a radius and a volume.
- Use the 1 : 2 : 3 relation to compare volumes.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the sphere and hemisphere formulas (questions 1 to 9).
- Criterion B. Investigating patterns: the two figures. Ask students to say the number of circles the peel covers for three different radii, and to find the 1 : 2 : 3 ratio for three different radii.
- Criterion C. Communicating: showing the radius first, then the cube or square, then the multiplication, as in the worked example.
- Criterion D. Applying mathematics in real-life contexts: an orange, a ball, a metal sphere, a bowl and balls in a tube (the hook, worked example and question 8).
The one thing to do with the figure
Make students predict the number of circles before they use the peel slider. Many say two or three; the surface of the sphere needs four. Ask: does the number depend on the radius? Move the radius and see that it does not.
In the second figure, ask for the sphere's volume as a fraction of the cylinder's before looking at the numbers. Then confirm that 2 out of 3 parts is 2/3.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: the orange peel. Collect predictions of the number of circles; do not settle them yet. |
| 10 min | First figure: cover the circles one at a time. Settle the hook. |
| 10 min | Key idea; second figure and the 1 : 2 : 3 ratio; the ball worked example; students finish the metal sphere in pairs. |
| 5 min | Question 1, the check. If many choose A or C, return to the squared radius. |
| 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 | 36π cm²; 54π cm³; 36. |
|---|---|
| Hook | The peel of an orange covers four circles of the orange's radius. |
| Worked example | 324π cm² and 972π cm³. |
| Faded example | 900π cm² and 4500π cm³. |
| 1. Which expression is the surface area? | B. A does not square; C uses the diameter; D is the volume. |
| 2. Surface area of a sphere | 144. 4 × 36. |
| 3. Volume of a sphere | 36. 4/3 × 27. |
| 4. Volume of a hemisphere | 144. Half of 288. |
| 5. Diameter given | 196. 4 × 49. |
| 6. A solid hemisphere's whole surface | 48. 32π + 16π = 3 × 16π. |
| 7. From surface area to volume | 2304. r = 12; 4/3 × 1728. |
| 8. Balls in a tube | 128. 384 − 256. |
| 9. Hemisphere: from surface to volume | A. 3r² = 27, r = 3, V = 2/3 × 27 = 18π. B gives the whole sphere; C takes r² as r; D squares instead of cubing. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Did not square, used the diameter, or gave the volume formula. | Ask: what shape is the unit of the area, and how many of them make the sphere? |
| 24, 36 or 576 in question 2 | Did not square, gave one circle, or used the diameter. | Write the area of one circle, then multiply it by 4. |
| 12, 108 or 9 in question 3 | Squared instead of cubing, forgot the 1/3, or used 1/3 not 4/3. | Write r³ as 3 × 3 × 3 first, and divide it by 3 before multiplying by 4. |
| 288, 72 or 24 in question 4 | Gave the whole sphere, a third of it, or squared the radius. | Say: half of the sphere. What is the sphere's volume? |
| 784, 56 or 49 in question 5 | Used the diameter, did not square, or gave one circle. | Cross out 14 and write the radius beside it first. |
| 32, 64 or 16 in question 6 | Gave only the curved part, the whole sphere, or only the flat circle. | Ask: how many faces does a solid hemisphere have, and what are they? |
| 3981312, 18432 or 192 in question 7 | Did not take the square root, did not divide by 4, or did not cube. | Work step by step: area to r², r² to r, r to volume. |
| 256, 384 or 640 in question 8 | Gave one volume, or added them. | Ask which volume is bigger, and what the question is asking for. |
| B, C or D in question 9 | Gave the whole sphere's volume, took r² as r, or squared instead of cubing. | Write 3πr² = 27π and solve for r first; then ask what fraction of 4/3 πr³ a hemisphere is. |
Questions worth asking
- Why is the surface area four circles and not three or five? The cylinder that just holds the sphere has a curved side of 2πr × 2r = 4πr², exactly the sphere's surface, a result known since ancient times. Listen for the peel experiment and for 4πr² as four times πr².
- Which is bigger: a sphere of radius 3, or a cylinder of radius 3 and height 6? The cylinder: 54π against 36π. The sphere is two thirds of it.
- What happens to the surface area and the volume when the radius doubles? Four times the area and eight times the volume, because r² and r³.
- Is the surface of a hemisphere half of the sphere's? Only the curved part. A solid hemisphere has an extra flat circle: 3πr², not 2πr².
Differentiation
A way in for anyone stuck
- Give a card with the two sphere formulas, a hemisphere row, and a worked line with r = 2 on it.
- Provide a list of cubes: 1, 8, 27, 64, 125, 216 for r up to 6.
- Use a real ball and a string to show the four circles: wrap and compare, or peel an orange together.
Stretch
- Question 9, then: a sphere just fits inside a cube. What fraction of the cube does the sphere fill? (π/6, about 52 percent.) Or: what radius gives a solid hemisphere a whole surface of 75π cm²? (r = 5.)
- A sphere has the same numerical surface area and volume. What is its radius? (4πr² = 4/3 πr³ gives r = 3.)
Students learning in English as an additional language
“Sphere” is a ball shape; “hemisphere” is half of a ball (“hemi” means half). “Surface” is the outside skin; “volume” is the space inside. Have students say “four circles” every time they find the surface area of a sphere.
| Word | Plain meaning | Example |
|---|---|---|
| sphere | a perfectly round solid, like a ball | a football |
| hemisphere | half of a sphere | a bowl |
| radius | the distance from the centre to the surface | 9 cm |
| diameter | the distance across through the centre, twice the radius | 18 cm |
| surface area | the area of the outside of a solid | 324π cm² |
| volume | the space inside a solid | 972π cm³ |
| ratio | a comparison of amounts, such as 1 : 2 : 3 | cone : sphere : cylinder |
Sentence starters for explaining aloud or in writing:
- The radius is …, so r squared is … and the area is …
- The volume of the sphere is 4/3 of …, which is …
- The cone, sphere and cylinder are in the ratio … because …
Two things not to say
“The sphere formula is 4/3 πr² and the area is 4πr³.” Area has the square and volume the cube: the units say which (cm² and cm³).
“A hemisphere is half a sphere, so halve the area.” Halve the curved part, and add the flat circle for a solid.
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.
- AreaA sphere has radius 5 cm. Find its surface area in terms of π.
100π cm². - VolumeA sphere has radius 6 cm. Find its volume in terms of π.
288π cm³.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA hemispherical bowl has an inside radius of 9 cm. How much does it hold, in terms of π?
2/3 × 729 = 486π cm³. - Earlier work (Lesson 1)A cylinder has radius 4 cm and height 5 cm. Find its volume in terms of π.
80π cm³.
3Challenge
Reasoning, generalizing and working backwards.
- Working backA sphere has volume 36π cm³. Find its radius, and then its surface area.
r³ = 27, so r = 3; area 36π cm². - ReasoningA sphere's radius is tripled. By what factor do its surface area and volume grow?
The area by 9 and the volume by 27.
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? Peel an orange and trace circles of its radius on paper: the peel covers four, as the figure shows.