The shape of the lesson
By the end of the lesson, learners can
- Circumference and area: In the lab, change the radius and press Roll it to see the circumference unroll into a straight line just over 3 diameters long.
- Arcs and sectors: In the lab, drag the red point to change the sector angle and watch the fraction \(\theta/360\) change.
- Any angle, and major sectors: Extended candidates meet any sector angle, the larger (major) sector, and questions that give the arc or area and ask for the angle or radius.
- Given: \(C = 2\pi r\) and \(A = \pi r^2\) (List of formulas).
- Radius: halve a diameter before using the formulas.
- Fraction: sector angle \(\div 360\).
- Arc and sector: fraction \(\times 2\pi r\) and fraction \(\times \pi r^2\).
- Perimeter of a sector: arc + 2 radii.
- Exact: "in terms of \(\pi\)" means keep \(\pi\) as a symbol.
One picture every idea returns to
Circumference \(C = 2\pi r\) and area \(A = \pi r^2\) are given in the List of formulas. For a sector with angle \(\theta\): arc length \(= \dfrac{\theta}{360} \times 2\pi r\) and sector area \(= \dfrac{\theta}{360} \times \pi r^2\). These two are NOT printed: build them from the fraction of the circle.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, A slice of pi.. Circumference and area of a circle are on the formula page. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Circumference and area | In the lab, change the radius and press Roll it to see the circumference unroll into a straight line just over 3 diameters long. Project tab 3 of the lab (Area and perimeter lab, circles) and let a student drive it while the class predicts. Model the worked example From the radius on the board, one line at a time. Your turn: A circle has circumference 50 cm. Find its radius. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Arcs and sectors | In the lab, drag the red point to change the sector angle and watch the fraction \(\theta/360\) change. Project tab 4 of the lab (Area and perimeter lab, arcs and sectors) and let a student drive it while the class predicts. Model the worked example Arc length and sector area on the board, one line at a time. Your turn: Find the perimeter of a semicircle of diameter 12 cm. | Pairs |
| 21 to 30 min | Teach and check: Any angle, and major sectors | Extended candidates meet any sector angle, the larger (major) sector, and questions that give the arc or area and ask for the angle or radius. Model the worked example Major sector (Extended) on the board, one line at a time. | Rally Coach |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 16 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Any angle, and major sectors, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Rally Coach
Pairs share one worksheet. One partner solves the first question aloud while the other coaches, checking each line and offering a prompt rather than the answer; they swap roles for the next question.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Putting the diameter into \(\pi r^2\): a circle of diameter 10 cm has area \(25\pi\), not \(100\pi\). | Halve the diameter before you square. |
| The perimeter of a sector is NOT just the arc. | Add the two radii (for a semicircle, the diameter). Leaving them out is the examiner's favourite lost mark. |
| When asked for a major sector, use \(360 - \theta\), not \(\theta\). | Read which region is shaded before you start. |
Circles, arcs and sectors: worksheet
6 questions, 16 marks. Written in the style of the papers; no past paper question is reproduced.
A circle has radius 5.2 cm. Calculate (a) its circumference (b) its area.
Without a calculator, find (a) the arc length (b) the area of a sector of radius 6 cm and angle 120°. Give your answers in terms of \(\pi\).
The circumference of a circle is 40 cm. Calculate its area.
(Extended) A major sector of a circle of radius 10 cm has angle 252°. Calculate (a) its area (b) its perimeter.
(Extended) A sector of a circle of radius 7 cm has area \(30\text{ cm}^2\).
Calculate (a) the sector angle (b) the arc length.
A bicycle wheel has diameter 66 cm. The bicycle travels 1 km in a straight line.
Calculate the number of complete turns the wheel makes.
Answer key and marking notes
- (a) \(2\pi \times 5.2 = 32.67... = 32.7\) cm
- (b) \(\pi \times 5.2^2 = 84.94... = 84.9\text{ cm}^2\)
- Fraction \(= \dfrac{120}{360} = \dfrac{1}{3}\).
- (a) \(\dfrac{1}{3} \times 12\pi = 4\pi\) cm
- (b) \(\dfrac{1}{3} \times 36\pi = 12\pi\text{ cm}^2\)
- \(2\pi r = 40\), so \(r = \dfrac{40}{2\pi} = 6.366...\) cm.
- Area \(= \pi \times 6.366...^2 = 127.3...\)
- Area \(= 127\text{ cm}^2\) (keep the unrounded radius in the calculator).
- Fraction \(= \dfrac{252}{360} = 0.7\).
- (a) Area \(= 0.7 \times 100\pi = 70\pi = 219.9... = 220\text{ cm}^2\)
- (b) Arc \(= 0.7 \times 20\pi = 14\pi = 43.98...\) cm; perimeter \(= 43.98... + 20 = 64.0\) cm
- (a) \(\dfrac{\theta}{360} \times 49\pi = 30\), so \(\theta = \dfrac{30 \times 360}{49\pi} = 70.15... = 70.2\)°
- (b) Arc \(= \dfrac{70.15...}{360} \times 14\pi = 8.571... = 8.57\) cm
- One turn moves the bicycle one circumference: \(\pi \times 66 = 207.3...\) cm.
- 1 km = 100 000 cm.
- \(100\,000 \div 207.3... = 482.28...\), so 482 complete turns.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
A circular table top has diameter 14 cm. Find its area, correct to 3 significant figures.
- A\(154\text{ cm}^2\)
- B\(616\text{ cm}^2\)
- C\(44.0\text{ cm}^2\)
- D\(88.0\text{ cm}^2\)
A circle has circumference 75 cm. Find its diameter, correct to 3 significant figures.
- A23.9 cm
- B11.9 cm
- C4.89 cm
- D236 cm
A sector has radius 12 cm and angle 30°. Find its area, in terms of \(\pi\).
- A\(12\pi\text{ cm}^2\)
- B\(2\pi\text{ cm}^2\)
- C\(144\pi\text{ cm}^2\)
- D\(24\pi\text{ cm}^2\)
Find the perimeter of a semicircle of diameter 9 cm, correct to 3 significant figures.
- A23.1 cm
- B14.1 cm
- C37.3 cm
- D18.6 cm
The diagram shows a circle of radius 10 cm. The minor sector has angle 72°. Find the area of the shaded major sector, correct to 3 significant figures.
- A\(251\text{ cm}^2\)
- B\(62.8\text{ cm}^2\)
- C\(50.3\text{ cm}^2\)
- D\(314\text{ cm}^2\)
A sector of a circle of radius 9 cm has area \(60\text{ cm}^2\). Find the sector angle, correct to 1 decimal place.
- A84.9°
- B382.0°
- C266.7°
- D42.4°
Quiz key
\(r = 14 \div 2 = 7\) cm, so \(A = \pi \times 7^2 = 49\pi = 153.9... = 154\text{ cm}^2\). The tempting 616 puts the diameter into the formula, \(\pi \times 14^2\). 44.0 is the circumference \(2\pi \times 7\), and 88.0 is \(2\pi \times 14\).
\(\pi d = 75\), so \(d = 75 \div \pi = 23.87... = 23.9\) cm. The tempting 11.9 cm is the radius \(75 \div 2\pi\), not the diameter. 4.89 cm treats 75 as an area (\(\sqrt{75 \div \pi}\)), and 236 cm multiplies by \(\pi\) instead of dividing.
Fraction \(= \dfrac{30}{360} = \dfrac{1}{12}\). Area \(= \dfrac{1}{12} \times \pi \times 12^2 = \dfrac{144\pi}{12} = 12\pi\text{ cm}^2\). The tempting \(2\pi\) is the arc length \(\dfrac{1}{12} \times 2\pi \times 12\). \(144\pi\) is the whole circle, and \(24\pi\) mixes the two formulas, \(\dfrac{1}{12} \times 2\pi \times 12^2\).
Curved edge \(= \tfrac{1}{2} \times 2\pi \times 4.5 = 4.5\pi = 14.13...\) cm. The straight edge is the diameter, 9 cm. Perimeter \(= 14.13... + 9 = 23.1\) cm. The tempting 14.1 cm is the arc alone. 37.3 cm uses 9 as the radius, and 18.6 cm adds a radius instead of the diameter.
Major angle \(= 360 - 72 = 288\)°. Area \(= \dfrac{288}{360} \times \pi \times 10^2 = 80\pi = 251.3... = 251\text{ cm}^2\). The tempting 62.8 is the MINOR sector, \(\dfrac{72}{360} \times 100\pi = 20\pi\). 50.3 is the major arc length (\(16\pi\)), and 314 is the whole circle.
\(\dfrac{\theta}{360} \times 81\pi = 60\), so \(\theta = \dfrac{60 \times 360}{81\pi} = 84.88... = 84.9\)°. The tempting 382.0° uses the arc formula \(2\pi r\) instead of \(\pi r^2\); an angle over 360° is impossible for a sector. 266.7° leaves out \(\pi\), and 42.4° doubles the area formula to \(2\pi r^2\).
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.