At a glance. One lesson of about 60 minutes. It assumes Pythagoras and the area of a circle. Students find slant heights, prove the curved area from the sector, find the sector angle, total surface areas of cones and pyramids, and work backwards from an area to a height.
Calculator: GDC optional. A GDC can store an unrounded slant height, which protects the final digit, but the lesson works by hand and keeps answers exact as multiples of π until the last step.
What students will learn
Know
- s² = r² + h² for a cone.
- Curved surface πrs, total πr² + πrs.
- A pyramid face's slant height comes from the vertical height and half of the base edge.
Understand
- The curved surface is a sector whose arc is the rim, so its fraction of the circle is r ÷ s.
- The same triangle appears in every question: choose which two sides are known.
- Working backwards uses the same relationships in the other order.
Do
- Find a slant height by Pythagoras.
- Explain why the curved area is πrs, and find the angle of the sector.
- Find the total surface area of a cone or a pyramid, and recover 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: slant height and surface area (questions 2 to 8).
- Criterion B. Investigating patterns: the what-if, how the sector angle changes as the height grows.
- Criterion C. Communicating: the proof in the key card, and a labelled triangle with each answer.
- Criterion D. Applying mathematics in real-life contexts: the lampshade, the cup and the hat, where the right measurement is the slanting one.
The one thing to do with the figure
Ask for the angle before showing it. Set r = 5 and h = 10 and ask for a prediction of the sector angle before the figure's number is read. Then raise h and ask which way the angle moves, and why.
Finish with the extreme settings: a very flat cone is nearly a full circle, a very tall one a thin slice.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the hat. Collect predictions for which cone needs the bigger slice. |
| 12 min | The figure with predictions; the key idea and the proof; 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; if C or D, to the rim and the arc. |
| 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 | 2.5 cm; 16π cm²; 1/6. |
|---|---|
| Hook | The tall, thin cone: its sector is the thinner slice, because r ÷ s is smaller. The short, wide cone uses the bigger slice. |
| Worked examples | 144π ≈ 452.4 cm²; 96 cm²; r = 15 cm and 375π ≈ 1178.1 cm². |
| Faded example | 154.3°. |
| 1. The sector as a fraction of a circle | B. A uses the vertical height, C is h ÷ s, D is upside down. |
| 2. Slant height by Pythagoras | 29. √(400 + 441). |
| 3. Slant height of a frustum | 37. √(35² + 12²). |
| 4. Total surface area of a cone | 678.6. 216π. |
| 5. Angle of the sector | 90. 360 × 3 ÷ 12. |
| 6. Working back to the height | 12. s = 13 first. |
| 7. A cone with a slant height tied to its radius | 4. 3r² = 48. |
| 8. Pyramid height from its surface | 24. s = 25, then √(625 − 49). |
| 9. How the curved area changes | C. πrs is proportional to r when s is fixed. A squares, B subtracts, D ignores r. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Used the vertical height, used h ÷ s, or inverted the fraction. | Find s from the triangle, then ask what fraction of the circle's edge the rim is. |
| 41 or 420 in question 2 | Added the sides, or multiplied them. | Square, add, square-root. |
| 41.3 or 47 in question 3 | Used the bottom radius alone, or added the legs. | Mark the difference of the radii on the sketch. |
| 424.1, 848.2 or 216 in question 4 | Left out the base, doubled the curved area, or dropped π. | Write the base and curved parts separately. |
| 1440, 45 or 18.8 in question 5 | Inverted the fraction, halved it, or found a length. | Say what fraction of the circle the rim is. |
| 13 or 8 in question 6 | Stopped at the slant height, or subtracted instead of using Pythagoras. | Ask which side the question wants. |
| 16 or 6.9 in question 7 | Stopped early, or left out the 3. | Write s as 3r first and expand. |
| 25, 18 or 20.7 in question 8 | Stopped at the slant height, subtracted, or used the whole base. | Halve the base before the triangle. |
| A, B or D in question 9 | Squared the radius, subtracted the radii, or thought the slant height fixes the area. | Write the area as π × r × s with s fixed. |
Questions worth asking
- Why is the fraction of the circle r ÷ s? The arc of the sector is the rim, 2πr, and the whole circle's edge is 2πs.
- Can the sector angle be more than 180°? Yes, when r is more than half of s, which is a flat, wide cone.
- Why does a pyramid face need its own slant height? The face slopes at a different angle from the sloping edge, and its triangle uses half the base edge.
- What changes if the pyramid's base is a rectangle? Two pairs of faces have two different slant heights.
Differentiation
A way in for anyone stuck
- Give a paper circle and let students cut a sector, roll it, and see the rim and the slanting edge.
- Provide a labelled triangle template with r, h and s and two blanks to fill.
- Keep the numbers in a list of Pythagorean triples (3, 4, 5; 5, 12, 13; 8, 15, 17) for the first questions.
Stretch
- A cone's curved surface is a sector of angle 120°. Show that its slant height is three times its radius, and find the ratio of its curved surface to its base area.
- Show that a cone of radius r and height h has a sector angle of 360r ÷ √(r² + h²), and find h in terms of r when the angle is 180°.
Students learning in English as an additional language
“Slant height” is the length along the sloping surface. “Vertical height” is straight up from the base. A “sector” is a slice of a circle. Practise saying “slant” while tracing the slope with a finger.
| Word | Plain meaning | Example |
|---|---|---|
| slant height | the length along the sloping surface, from the rim to the apex | s = 17 for r = 8 and h = 15 |
| vertical height | the straight-up distance from the apex to the centre of the base | the 36 in a cone with s = 39 and r = 15 |
| sector | a slice of a circle between two radii | a quarter circle |
| curved surface | the part of a cone's surface that is not the base | πrs |
| apex | the point of a cone or pyramid opposite its base | the tip of a hat |
Sentence starters for explaining aloud or in writing:
- The slant height is found from the right-angled triangle with sides …
- The sector is the fraction … of the whole circle because …
- I halved the base because …
Two things not to say
“Height is height.” Always ask: vertical or slant?
“Just use the formula card.” Ask where πrs comes from, and let the sector answer.
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.
- Slant heightA cone has radius 9 cm and vertical height 12 cm. Find its slant height.
15 cm. - Curved areaA cone has radius 2 cm and slant height 11 cm. Find its curved surface area in terms of π.
22π cm².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA tent in the Gobi is a square pyramid with base side 10 m and vertical height 12 m. Find the area of canvas in the four sides.
260 m², from four faces of ½ × 10 × 13. - Earlier work (Year 9)The area of a circle is 50.3 cm². Find its radius to one decimal place.
4.0 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA cone of radius 5 cm has a curved area equal to twice its base area. Find the slant height.
10 cm. - ProofShow that the sector angle of any cone is at most 360° and equals 360° only when the height is 0.
The angle is 360r ÷ s and s ≥ r, with s = r only when h = 0.
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.