At a glance. One lesson of about 60 minutes. It follows lessons 1 to 3 and assumes Pythagoras' theorem. Students find the volume, slant height, curved surface area and whole surface of a cone, in terms of π and to one decimal place, and work back to a height.
Calculator: scientific calculator. The π key is needed for the one-decimal-place answers. The slant heights are whole numbers (3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25; 15, 20, 25), so the square roots are exact and the lesson is not about buttons.
What students will learn
Know
- The volume of a cone is ⅓πr²h.
- The slant height satisfies l² = r² + h².
- The curved surface area is πrl, and the whole surface is πr² + πrl.
Understand
- The cone holds a third of the cylinder with the same base and height, as the pouring shows.
- The curved side unrolls into part of a circle of radius l, so its area is πrl.
- The height and the slant height are different lengths with different jobs.
Do
- Find the volume of a cone, in terms of π and to one decimal place.
- Find the slant height and the curved or whole surface area.
- Work back to a height from a volume and a radius.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the cone formulas and Pythagoras (questions 1 to 9).
- Criterion B. Investigating patterns: the first figure. Ask students to predict the level of water after one, two and three cone-fuls, then to link the pattern to the formula.
- Criterion C. Communicating: sketching the cone with its three lengths, writing l before using it, as in the worked example.
- Criterion D. Applying mathematics in real-life contexts: a chestnut cone, a roof, a conical hat and a grain heap (the hook, the worked example, questions 7 and 8).
The one thing to do with the figure
Pour before you state. Ask for the number of cone-fuls that fill the cylinder before anyone touches the sliders, then pour with the third slider. Change the radius and height to show it stays a third.
In the second figure, set the radius 12, height 5 cone and the radius 5, height 12 cone one after the other: the slant height is 13 for both, but the volumes are different. Ask what is the same and what is not.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: the vendor's cone and tub. Collect predictions of the number of cone-fuls. |
| 10 min | First figure: pour one, two, three cone-fuls. Settle the hook. |
| 10 min | Key idea; second figure and the right-angled triangle; the worked example; students finish the 12 and 16 cone in pairs. |
| 5 min | Question 1, the check. If many choose B, return to the height against the slant height. |
| 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 cm³; 13 cm; 8π cm. |
|---|---|
| Hook | Three cones fill the tub; one cone is a third. |
| Worked example | V = 320π ≈ 1005.3 cm³; l = 17 cm; curved surface 136π ≈ 427.3 cm². |
| Faded example | l = 20 cm; 240π ≈ 754.0 cm². |
| 1. Which expression is the volume? | A. B uses the slant height; C is the cylinder; D does not square the radius. |
| 2. Volume of a cone in terms of π | 21. ⅓ × 9 × 7. |
| 3. Slant height from radius and height | 10. √(64 + 36). |
| 4. Curved surface area | 175. π × 7 × 25. |
| 5. Whole surface to one decimal place | 1131.0. 360π = 1130.97… |
| 6. Working back to the height | 18. 3 × 96 ÷ 16. |
| 7. Leaves on a conical hat | 1570.8. 500π = 1570.79… |
| 8. A conical heap | 42.4. 13.5π = 42.41… |
| 9. Curved surface of a tall cone | B. A uses the height; C is the volume; D includes the base. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Used the slant height, the cylinder, or did not square the radius. | Ask: which of the lengths is straight up inside the cone? Which is on the surface? |
| 63, 7 or 31.5 in question 2 | Gave the cylinder, did not square, or used a half. | Pour the cone into the cylinder in the figure: how many fill it? |
| 14, 28 or 100 in question 3 | Added, subtracted the squares, or left out the square root. | Draw the triangle and mark the hypotenuse first. |
| 1225, 350 or 32 in question 4 | Squared the radius, used the cylinder's formula, or added. | Ask where the π r l comes from: the flat sector's area. |
| 816.8, 1068.1 or 314.2 in question 5 | Left out the base, used the height as the slant height, or left out the curved side. | List the two parts, base and curved side, and tick each. |
| 6, 72 or 4608 in question 6 | Forgot to undo the third, did not square the radius, or multiplied. | Say in words: a third of the base times the height is 96π; undo it step by step. |
| 942.5, 3141.6 or 500 in question 7 | Used the height, a cylinder's formula, or left out π. | Ask for the slant height first and label it on a sketch. |
| 127.2, 169.6 or 9.4 in question 8 | Forgot the third, used the diameter, or did not square. | Ask for the radius and the base area separately before the volume. |
| A, C or D in question 9 | Used the height, the volume, or included the base. | Ask: what is the question asking for, and which length lies along the surface? |
Questions worth asking
- Why is the cone exactly a third and not almost a third? It can be shown, but not by a simple picture; listen for the experiment of three cone-fuls and the same one-third rule as the pyramid.
- Where does πrl come from? Open out the cone with the button in the second figure: part of a circle of radius l and an edge of 2πr. The area of the part is ½ × l × 2πr.
- Can two different cones have the same slant height? Yes: 5, 12, 13 and 12, 5, 13 both have l = 13, with different volumes.
- How does a cone compare with a pyramid? Both are a third of the prism or cylinder with the same base and height; a cone is a pyramid with a circle base.
Differentiation
A way in for anyone stuck
- Provide a triangle card with r, h and l labelled, and the three rules written under it.
- Give the common triples to look for: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25.
- Use a real cone and a cylinder of the same size and pour rice between them for questions 1 and 2.
Stretch
- Question 9, then: find the whole surface area and the volume of the cone in terms of π. (450π cm² and 1080π cm³.)
- A cone has radius r and slant height 3r. Show that its curved surface area is 3πr² and find its height in terms of r. (h = r√8.)
Students learning in English as an additional language
“Slant” means leaning. “Apex” is the point at the top. The word “curved” goes with the surface of the cone and “base” with the circle at the bottom. Have students say “height, inside” and “slant height, on the surface”.
| Word | Plain meaning | Example |
|---|---|---|
| cone | a solid with a circular base and one point at the top | a paper cup for chestnuts |
| apex | the point at the top of a cone | the tip of a party hat |
| perpendicular height | the height straight up from the base to the apex | 12 cm |
| slant height | the length from the apex to the edge of the base, along the surface | 13 cm |
| curved surface | the part of a cone that is not the circular base | the leaf covering of a hat |
| sector | a slice of a circle, like a piece of pie | the flat shape of a cone's side |
Sentence starters for explaining aloud or in writing:
- The cone holds a third of the cylinder because …
- The slant height is … because …
- The curved surface area is π × … × …, which is …
Two things not to say
“A cone is half a cylinder.” It is a third; the pouring shows it.
“Use whichever height is given.” The volume needs the perpendicular height; the curved surface needs the slant height.
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.
- VolumeA cone has radius 3 cm and height 4 cm. Find its volume in terms of π.
12π cm³. - SlantA cone has radius 9 cm and height 12 cm. Find its slant height.
15 cm.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA traffic cone is a cone with radius 15 cm and slant height 39 cm. Find its curved surface area to one decimal place.
π × 15 × 39 = 585π = 1837.8 cm². - Earlier work (Year 8)A circle has radius 6 cm. Find its area in terms of π.
36π cm².
3Challenge
Reasoning, generalizing and working backwards.
- Working backA cone has curved surface area 65π cm² and radius 5 cm. Find its slant height and its height.
l = 13 cm; h = 12 cm. - ReasoningA cone and a cylinder have the same base and the same volume. How do their heights compare?
The cone is 3 times as tall.
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? A paper cone, a paper cylinder of the same size and a cup of rice show the third. Cut a cone's side open to see the sector.