At a glance. One lesson of about 60 minutes. It assumes the circle formulae from Year 8. Students find an arc, a sector and a perimeter, work back to a radius or an angle, give answers in terms of π, and find the area of a segment.
Calculator: GDC optional. A calculator is used for decimals and the sine, so the label is GDC optional. A scientific calculator does the same work, and the closed section covers keeping π in the key sequence.
What students will learn
Know
- arc = (t/360) × 2πr and sector area = (t/360) × πr².
- Perimeter of a sector = arc + 2r.
- Segment = sector − ½r² sin t.
Understand
- A sector is a fraction of a circle, so every length and area is that fraction of the circle's own.
- Length scales with the radius and area with its square.
Do
- Find an arc, area and perimeter of a sector.
- Work back to a radius or an angle.
- Find the area of a segment.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using the arc, area and perimeter formulae and the area of a segment (questions 1 to 9).
- Criterion B. Investigating patterns: the first figure. Students say what doubles and what quadruples when the radius or angle is doubled.
- Criterion C. Communicating: an answer in terms of π and a decimal, with the fraction written down (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the two cakes and the flower bed.
The one thing to do with the figure
Make students predict, then reveal. Before touching the first figure, ask: if the radius doubles, what happens to the arc and what to the area? Write it down, then set the radius from 6 to 12 and compare.
In the second figure, set the angle to 180° and ask why the segment equals the sector there.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the two cakes. Collect votes and reasons. |
| 12 min | First figure; key idea; settle the hook; the warning card. |
| 10 min | Second figure; worked examples 1 to 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or B, return to how length and area scale. |
| 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 | 10π cm; 113.1 cm²; 32 cm². |
|---|---|
| Worked examples | 7π = 22.0 cm and 42π = 131.9 cm²; 9.1 cm²; the proof gives ½ × arc × r. |
| Faded example | 45π ≈ 141.4 cm². |
| Something to think about | The angle is divided by 4 (80° to 20°) and the arc halves (8π/3 to 4π/3). |
| 1. Scaling an arc and an area | D. A treats area like length; B ignores the halved angle; C swaps the two results. |
| 2. Arc length | 14.1 cm. (54 ÷ 360) × 30π. |
| 3. Area of a sector with a reflex angle | 219.9 cm². 0.7 × 100π. |
| 4. Radius from an arc length | 15.3 cm. r = 7200 ÷ (150π). |
| 5. Perimeter of a sector | 32.3 cm. 18.33 + 14. |
| 6. Angle from an area | 72°. 40 ÷ (64π) × 360. |
| 7. Area of a segment from its chord | 45 cm². 2 sin−1 0.8 ≈ 106.3°; 92.7 − 48. |
| 8. Area between two arcs | 86.4 m². (75/360) × 132π. |
| 9. Working back from an arc length in terms of pi | C. A uses the arc as the whole circle; B forgets the quarter; D gives the circumference's coefficient. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Treated area like length, ignored the halved angle, or swapped the results. | Ask: what does doubling r do to a length, and to an area? What does halving the angle do to each? |
| 7.1 or 106 in question 2 | Left out the 2, or found the area. | Write the circumference first, 2πr, then take the fraction of it. |
| 94.2 or 22 in question 3 | Used the other part of the circle, or did not square the radius. | Say: which part of the circle is shaded, and what is its angle? |
| 30.6 or 5.5 in question 4 | Left out the 2, or took a square root. | Ask: does r appear once or twice in an arc? |
| 18.3 or 25.3 in question 5 | Gave the arc only, or added one radius. | Trace the outline with a finger: how many straight edges? |
| 573 or 286 in question 6 | Did not square r, or divided by the circumference. | Find the circle's whole area first, then the fraction 40 over it. |
| 6 or 93 in question 7 | Used the half-angle at the centre, or gave the sector. | Join the centre to both ends of the chord: which angle is the sector's angle, and what is removed from the sector? |
| 23.6 or 128.3 in question 8 | Squared the difference, or gave the outer sector. | Draw both sectors with the same angle and subtract their areas. |
| A, B or D in question 9 | Used the arc as a whole circumference, forgot the quarter, or gave the circumference. | Write the circumference as an equation first, 20π = 2πr. |
Questions worth asking
- Why does the area quadruple when the radius doubles, but the arc only doubles? Listen for “area has two lengths multiplied, length has one”.
- Why does the segment equal the sector at 180°? Listen for “the triangle has no area when the chord is a diameter”.
- Does the sector formula work for angles over 180°? Listen for “yes, it is still a fraction of 360°”.
- Why is a sector like a triangle with base the arc and height the radius? Listen for “the area comes out as half arc times radius”.
Differentiation
A way in for anyone stuck
- Give a fraction wheel with the angle as a fraction of 360 for the first four questions.
- Provide a formula card with arc, area and perimeter, each with its fraction first.
- Restrict to questions 1 to 4 at first, with friendly angles such as 90°, 120° and 180°.
Stretch
- Show that for a segment with angle t in radians, the area is ½r²(t − sin t), and say what happens for very small t.
- A sector of angle 120° is rolled up into a cone. Find the radius of the cone's base in terms of the sector's radius.
Students learning in English as an additional language
“Arc” is a curved piece of the circle's edge. “Sector” is a slice like a pizza slice. “Segment” is the piece between a chord and the arc. Use a drawn circle and colour each in turn.
| Word | Plain meaning | Example |
|---|---|---|
| arc | a part of the circumference of a circle | the crust of a pizza slice |
| sector | the region between two radii and an arc | a slice of cake |
| segment | the region between a chord and the arc it cuts off | the cap cut off a circle by a straight line |
| chord | a straight line joining two points on a circle | a bowstring |
| perimeter | the total length around the outside of a shape | the fence around a garden |
Sentence starters for explaining aloud or in writing:
- The fraction of the circle is … because the angle is …
- I use 2πr for … and πr² for …
- The segment is the … take away the …
Two things not to say
“Just plug in the formula.” Ask: which fraction of the circle, and a length or an area?
“The perimeter is the arc.” Include the straight edges.
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.
- ArcA sector has radius 6 cm and angle 60°. Find its arc, in terms of π.
2π cm. - Earlier work (Year 8)Find the circumference of a circle of diameter 9 cm, in terms of π.
9π cm.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- AreaA sector has radius 10 cm and angle 36°. Find its area in terms of π.
10π cm². - BackA sector's arc is 6π cm and its angle is 90°. Find the radius.
12 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why doubling the angle of a sector doubles its area but doubling the radius quadruples it.
Area is proportional to the angle and to the radius squared. - OpenA sector has radius 9 cm and a perimeter of 30 cm. Find its angle at the centre, to the nearest degree.
76°: the arc is 30 − 18 = 12, and 12 ÷ (18π) × 360 ≈ 76.4.
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 plates cut into sectors let students compare an arc with a string laid along it. The With a GDC section is optional.