At a glance. One lesson of about 60 minutes. It assumes Pythagoras and the circle area from Year 9. Students find slant heights, then the curved and total surface area of cones and the total surface area of square-based pyramids, leaving π in exact answers.
Calculator: GDC optional. The lesson is done by hand with π left in the answer. The optional section shows GDC users how to enter the whole expression once and not round in the middle.
What students will learn
Know
- Slant height s = √(r² + h²) for a cone; l = √(h² + (b/2)²) for a face of a square-based pyramid.
- Cone: curved surface πrs, total πr² + πrs.
- Square-based pyramid: total b² + 2bl.
Understand
- The slant height is the hypotenuse of a triangle inside the solid, so it is always longer than the vertical height.
- The surface of a solid is made of shapes whose measurements are found on the surface, not along its axis.
Do
- Find a slant height by Pythagoras, in either direction.
- Find the curved and total surface area of a cone and the total surface area of a pyramid.
- Work back from a total surface area to 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, curved and total surface area of cones and pyramids (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students find which radius and height pairs give a whole-number slant height and describe how s compares with h.
- Criterion C. Communicating: drawing the right-angled triangle and labelling its legs and hypotenuse in every working (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the woven hat, the tent and the party hat.
The one thing to do with the figure
Settle the hook with the triangle, not a rule. Ask students to set the sliders to a radius and height they choose, then to say which side of the triangle is the strip of straw. Ask: why can the slant never be shorter than the height?
In the second figure, ask where the foot of the slant height l sits on the base. It is the middle of a base edge, so the leg on the base is b/2, not b.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the hat. Collect votes and reasons for the strip length. |
| 12 min | First figure; key idea; settle the hook; the warning card. |
| 8 min | Second figure; the three worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to the triangle. |
| 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 | 25 cm; 16π cm²; 35 cm². |
|---|---|
| Hook | 20 cm: √(12² + 16²) = √400. |
| Worked examples | 600π cm²; 1200 cm²; 55.0 cm². |
| Faded example | 896 cm². |
| 1. Finding the slant height | B. A adds the legs; C subtracts them; D multiplies them. |
| 2. Curved surface of a cone | 60. s = 10, so πrs = 60π. |
| 3. Slant height of a pyramid face | 13. √(144 + 25). |
| 4. Total surface area of a pyramid | 360. 100 + 260. |
| 5. Canvas for a conical tent | 113.1 m². 36π. |
| 6. Working back to a vertical height | 15. √(289 − 64). |
| 7. Working back to a slant height | 8. 7s = 56. |
| 8. A paper party hat | 817 cm². 260π. |
| 9. Total surface area of a cone | D. A uses h instead of s; B leaves out the base; C does both. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Added, subtracted or multiplied the legs. | Draw the triangle and write s² = r² + h² before any number. |
| 48 or 36 in question 2 | Used the vertical height, or gave the base area. | Ask: which length runs along the outside of the cone? |
| 17 or 22 in question 3 | Added the legs, or used the whole base edge. | Mark the foot of the slant on the middle of a base edge. |
| 260 or 340 in question 4 | Left out the base, or used the vertical height for the faces. | List the five faces of the pyramid before calculating. |
| 163.4, 36 or 50.3 in question 5 | Added the floor, left out π, or gave the floor. | Ask whether the floor is covered by canvas. |
| 9 or 33 in question 6 | Subtracted or added lengths instead of squares. | Rearrange Pythagoras: h² = l² − (b/2)². |
| 15 or 56 in question 7 | Forgot the base, or forgot to divide by r. | Write 49π + 7πs = 105π and solve in two steps. |
| 754 or 1131 in question 8 | Used the vertical height, or added a base the hat does not have. | Ask which faces the paper actually covers. |
| A, B or C in question 9 | Used h for the curved surface, left out the base, or both. | Write the two parts of the surface on separate lines first. |
Questions worth asking
- Why is the slant height always longer than the vertical height? Listen for “it is the hypotenuse of a right-angled triangle, the longest side”.
- Why is half the base edge, not the whole edge, a leg of the pyramid's triangle? The slant of a face meets the base edge at its middle.
- What happens to s when r gets very small? It gets closer to h: the cone becomes a thin spike.
- Why does the cone's total surface need a base but the party hat's paper does not? The hat is open at the bottom; the question decides which faces are counted.
Differentiation
A way in for anyone stuck
- Give a cut-out cone net and a cut-out pyramid net and let students measure l and s along the paper.
- Provide a frame: “s² = ____² + ____² = ____, so s = ____.”
- Restrict to whole-number slant heights (a card of right-angled triples) before surds.
Stretch
- A square-based pyramid has slant height l and base edge b. Show that its total surface area is b(b + 2l).
- A cone's slant height is three times its radius. Write its vertical height in terms of r.
Students learning in English as an additional language
“Slant” means leaning, not straight up. “Vertical height” is straight up from the base. Say “s squared equals r squared plus h squared” aloud so the spoken form is clear, and point to the sides of the triangle as you say it.
| Word | Plain meaning | Example |
|---|---|---|
| slant height | the length of the sloping side from the apex to the edge of the base | the sloping edge of a roof, from the ridge to the gutter |
| vertical height | the distance straight up from the base to the apex | h in the cone |
| curved surface | the part of a cone's surface that is not its base | πrs |
| total surface area | the area of every face, including the base | curved surface plus base |
| apex | the point at the top of a cone or pyramid | the tip of the hat |
| hypotenuse | the longest side of a right-angled triangle, opposite the right angle | the side of length 5 in a 3, 4, 5 triangle |
Sentence starters for explaining aloud or in writing:
- The slant height is the hypotenuse because …
- The legs of the triangle are … and …
- The total surface area is made of …
Two things not to say
“Just use the height.” The surface needs the slant height; the height is a leg of the triangle.
“Memorize the formula.” Draw the triangle: the formula comes from it each time.
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 20 cm and vertical height 21 cm. Find its slant height.
29 cm. - Earlier work (Year 9)A cylinder has radius 5 cm and height 8 cm. Find its curved surface area in terms of π.
80π cm².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA pyramid-shaped roof in Kyoto has a square base of edge 18 m and vertical height 12 m. Find the area of its four sloping faces.
l = 15 m, so 2 × 18 × 15 = 540 m². - Working backA cone with radius 5 cm has curved surface 65π cm². Find its slant height and vertical height.
s = 13 cm and h = 12 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA cone's slant height is twice its radius. Show that its curved surface is twice its base area.
πr(2r) = 2πr², and the base is πr². - OpenFind a cone with radius more than 20 cm and whole-number vertical height whose slant height is also a whole number.
For example r = 24 and h = 7 give s = 25.
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? Paper cones and a ruler are enough: students measure the slant side along the paper and the vertical height with a set square. The With a GDC section is optional.