At a glance. One lesson of about 60 minutes. It assumes circumference and area of a circle. Students find arc length, sector area and perimeter, and work back to an angle or radius.
Calculator: GDC optional. Decimals of π need a calculator, so the label is GDC optional. Exact answers with π are encouraged, and the optional section is about the π key and bracketing.
What students will learn
Know
- Arc length = (t/360) × 2πr.
- Sector area = (t/360) × πr².
- Perimeter of a sector = arc + 2r.
Understand
- A sector's arc and area are the same fraction of the circle's circumference and area.
- Doubling the angle and halving the radius keeps the arc the same but halves the area.
Do
- Find an arc length, sector area and perimeter.
- Work back to an angle or a radius.
- Find the area of a band between two radii.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: arc length and sector area as fractions of a circle (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students find which angle and radius pairs give the same arc and how the perimeter splits into an arc and two radii.
- Criterion C. Communicating: the fraction written in simplest form, then the full expression, then the answer with units.
- Criterion D. Applying mathematics in real-life contexts: the baker's trays and the band between two radii.
The one thing to do with the figure
Make the fraction visible. In the first figure, ask students to set the angle to 90° and say what fraction of the circle is shaded. Then ask: what fraction of the area, and of the edge?
In the second figure, ask: when is the perimeter longer than twice the arc? (When the arc is shorter than 2r, for small angles.)
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the two slices. Collect votes. |
| 12 min | First figure; key idea; settle the hook with worked example 1; the warning card. |
| 10 min | Second figure; worked examples 2 and 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to which quantity is a length. |
| 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 | 44.0 cm; 36π cm²; 5/12. |
|---|---|
| Hook | The same: both arcs are 4π ≈ 12.6 cm. |
| Worked examples | 12.6 cm each; 15.7 cm and 70.7 cm²; 95.5°. |
| Faded example | 17.5π ≈ 55.0 cm². |
| Something to think about | About 245°: t = 360 − 360/π ≈ 245.4°. |
| 1. Which expression gives the arc | B. A is the area; C inverts the fraction; D leaves out the 2. |
| 2. Arc length | 14.7 cm. |
| 3. Sector area | 94.2 cm². |
| 4. Angle from an arc length | 79°. 78.8 rounded. |
| 5. Perimeter of a sector | 30.5 cm. |
| 6. Angle from an area | 71.6°. |
| 7. A band between two radii | 52.4 cm². |
| 8. Radius from an arc length | 11.9 cm. |
| 9. Exact area of a sector | C. A is the arc; B is the whole circle; D is the missing part. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Gave area, inverted the fraction, or used πr. | Ask: is the curved edge a length or an area, and what is the whole length round the circle? |
| 88 or 7.3 in question 2 | Gave the area, or used πr. | Write the circumference formula before using the fraction. |
| 12.6 or 188.5 in question 3 | Gave the arc, or put a 2 in the area. | Ask: which formula has r squared? |
| 158 in question 4 | Used πr. | Write the circumference in full, 2 × π × 8, before dividing. |
| 16.5 or 23.5 in question 5 | Left out both radii, or only one. | Trace the perimeter with a finger: how many straight sides? |
| 573 in question 6 | Used πr instead of πr². | Say: the area of the whole circle is π r squared. |
| 10.5 or 94.2 in question 7 | Squared the difference of radii, or stopped at the large sector. | Subtract the two sector areas. |
| 23.9 in question 8 | Used πr. | Write (1/3) × 2πr = 25 before rearranging. |
| A, B or D in question 9 | Gave the arc, the whole circle, or the missing part. | Ask: which fraction is 225 out of 360, and of what? |
Questions worth asking
- Why is the arc the same fraction of the circumference as the angle is of 360°? Listen for “equal angles at the centre cut off equal arcs, so the arc grows in proportion to the angle”.
- Why does doubling the angle and halving the radius keep the arc the same? Arc is proportional to both.
- Why can the perimeter of a sector be longer than its arc by exactly 2r? Two straight sides of length r are added.
- When is a sector's area numerically equal to half its arc times r? Always: area = ½ × arc × radius.
Differentiation
A way in for anyone stuck
- Give a pre-simplified fraction card for common angles: 30°, 45°, 60°, 90°, 120°, 180°, 270°.
- Provide a frame: “fraction = ____/360 = ____; whole = ____; part = fraction × whole”.
- Restrict to angles that divide 360 exactly (30°, 45°, 60°, 90°, 120°).
Stretch
- Show that the area of a sector is half the arc length times the radius.
- A sector of angle t is rolled into a cone. If the radius of the sector is r, find the base radius of the cone in terms of t and r.
Students learning in English as an additional language
“Arc” is a part of the edge of a circle. “Sector” is a slice, like a piece of pizza. “Perimeter” is the distance all the way round the outside. Hold up a paper circle and fold it to show a quarter and a half.
| Word | Plain meaning | Example |
|---|---|---|
| arc | a part of the curved edge of a circle | the edge of a slice of pizza |
| sector | a region between two radii and an arc | a slice of a round cake |
| circumference | the distance all the way round a circle | the length of a ring of string |
| central angle | the angle at the centre between two radii | the point of the slice |
| fraction of a circle | the part of 360° that an angle covers | 90° out of 360° is a quarter |
| perimeter | the length of the whole boundary of a shape | the fence round a field |
Sentence starters for explaining aloud or in writing:
- The fraction of the circle is … because the angle is … out of 360.
- The arc is that fraction of the …
- The perimeter is the arc plus …
Two things not to say
“Plug into the formula.” Ask: what fraction of the circle is it?
“Add the arc and the area.” One is a length and one is an area.
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.
- FractionWhat fraction of a circle is a sector of 45°?
1/8. - Earlier work (Year 8)Find the circumference of a circle with diameter 10 cm, in terms of π.
10π cm.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA clock's minute hand is 12 cm long. How far does its tip move in 20 minutes, to 1 decimal place?
25.1 cm. - Working backA sector with radius 10 cm has area 50 cm². Find its angle to 1 decimal place.
57.3°.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that area = ½ × arc length × radius for a sector.
Area = (t/360)πr² and arc = (t/360)2πr, so ½ × arc × r = (t/360)πr². - OpenFind a radius and an angle (a multiple of 10°) for which the arc is exactly 2π.
For example r = 9 and 40°, or r = 6 and 60°.
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 circles folded into twelfths give sectors to measure with string. The With a GDC section is optional.