At a glance. One lesson of about 60 minutes. Students explain why a circle's area is πr², use the radius (not the diameter), find a radius from an area, and describe how area grows as the radius grows.
Calculator: scientific calculator. Squaring decimals and multiplying by 3.14 is slow by hand. The calculator does the arithmetic; the estimate of three squares on the radius is the check.
What students will learn
Know
- A = πr², with r the radius.
- The radius is half of the diameter.
- Area is in square units, circumference in plain units.
Understand
- Sectors fit into a strip of length πr and height r, so the area is πr × r.
- Doubling the radius gives four times the area, because r is squared.
- A circle holds a little over three squares on its radius.
Do
- Find the area of a circle from its radius or diameter.
- Find a radius from an area.
- Find the area of a ring, or of a circle with a rectangle.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the formula, radius and diameter, and square units (questions 1 to 9).
- Criterion B. Investigating patterns: predict the area ratio for radii in the ratio 2, 3 and 1.5, test with the second figure, and generalize (the second figure and question 8).
- Criterion C. Communicating: write the radius on the diagram, show the squaring, and say why the answer is in square units (questions 3 and 6).
- Criterion D. Applying mathematics in real-life contexts: pizzas, ponds and tunnels, where the question decides what is added and what is taken away (questions 6 and 7 and the pizzas).
The one thing to do with the figure
Make the squares visible. In the second figure, draw attention to the dashed square on each radius. Ask how many of those squares fit into the circle. Students see a little over three, then check: 28.26 ÷ 9 = 3.14.
Ask for a prediction before every change to the radius: the area when the second radius is double, triple, one and a half times. Most say “double”, which is the doubling the lesson corrects.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | The pizzas and the prediction. First figure: build the strip. |
| 10 min | Key idea. Students cut a paper circle into 8 or 16 sectors and rearrange them. |
| 10 min | Second figure, worked examples, then the faded one in pairs. |
| 5 min | Question 1, the check. If many choose B, return to the common-mistake card. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 25; 62.8 cm; 60 cm². |
|---|---|
| Worked examples | 314 cm² and 706.5 cm², 2.25 times; 7.065 m²; 81π and 254.34 cm². |
| Faded example | 530.66 m². |
| 1. Area from the diameter | A. B squares the diameter, C is the circumference, D forgets to square. If many choose B, return to the common-mistake card and the second figure. |
| 2. Area from the radius | 113.04 m². 3.14 × 36 = 113.04. |
| 3. In terms of pi | 49. A = 49π cm². |
| 4. Area from the diameter | 153.86 m². 3.14 × 49 = 153.86. |
| 5. Missing radius | 5 cm. 78.5 ÷ 3.14 = 25 and 5 × 5 = 25. |
| 6. A ring | 263.76 m². 314 − 50.24 = 263.76. |
| 7. A rectangle and a semicircle | 32.13 m². 18 + 14.13 = 32.13. |
| 8. How the area grows | 6.25 times. 2.5 × 2.5 = 6.25. |
| 9. Circumference or area | A. B swaps the two formulas, C takes π × r as the circumference, D squares the diameter. If many choose B, ask which answer must be in square units. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B in question 1 | Squares the diameter. | Write the radius on the diagram before squaring. Ask which length goes in. |
| 452.16, 37.68 or 18.84 in question 2 | Uses the diameter, finds the circumference, or forgets to square. | Ask for the unit: square metres, so two lengths multiplied. |
| 14 or 7 in question 3 | Doubles the radius, or leaves it. | Write π × r × r and fill in r. |
| 615.44 or 43.96 in question 4 | Uses the diameter, or finds the circumference. | Ask what the radius is and show it on the diagram. |
| 25 or 12.5 in question 5 | Stops at r², or halves instead of taking a square root. | Ask which number times itself makes 25. |
| 113.04 or 364.24 in question 6 | Subtracts the radii, or adds the pond. | Colour the grass. Is the pond in the grass? |
| 46.26 or 74.52 in question 7 | Uses a whole circle, or the diameter as the radius. | Ask what fraction of a circle is there and what the radius is. |
| 2.5 or 12 in question 8 | Reports the ratio of the radii, or subtracts. | Work out both areas, then divide. |
| B, C or D in question 9 | Swaps the formulas, takes π × r as the circumference, or squares the diameter. | Ask which answer is a length and which an area, then estimate each: three diameters, three squares on the radius. |
Questions worth asking
- Why is the strip's height the radius? Each sector is as long as the radius, and the sectors stand up and down. Listen for “the sector's length”.
- Why is the strip's length half the circumference? Half of the sectors have their curved edge on the bottom and half on the top. Listen for “half on each side”.
- If the radius is doubled, why is the area not doubled? It grows in two directions: the strip is longer and higher. Listen for “both length and height”.
- Which is bigger, a 30 cm pizza or two 20 cm pizzas? Two 20 cm pizzas are 628 cm², and one 30 cm pizza is 706.5 cm². Listen for the squared growth.
Differentiation
A way in for anyone stuck
- Give a paper circle already divided into sectors, in two colours, to rearrange.
- Mark the radius in colour on every diagram before squaring.
- Ask for the estimate three squares on the radius before the calculator.
Stretch
- A circle has an area of 200.96 cm². Find its radius and its circumference. (r = 8 cm, C = 50.24 cm.)
- A square just fits a circle of radius 10 cm. What fraction of the square's area does the circle take? (314 ÷ 400 = 0.785.)
Students learning in English as an additional language
“Squared” has two meanings: multiplied by itself, and a square shape. Here both are true: r × r is the area of a square with side r. “Area” is how much flat space; say “square centimetres” aloud so the unit is heard.
| Word | Plain meaning | Example |
|---|---|---|
| area | the amount of flat space inside a shape | How much pizza there is |
| radius | a straight line from the centre of a circle to its edge | Half of the width of a pizza |
| squared | multiplied by itself | 6 squared is 36 |
| sector | a slice of a circle, like a slice of pizza | One of the eight slices |
| ring | the region between two circles with the same centre | The grass round a pond |
| semicircle | half of a circle | The top of an arched tunnel |
Sentence starters for explaining aloud or in writing:
- The diameter is … so the radius is …
- The area is π times … times …, because …
- If the radius is multiplied by …, the area is multiplied by …
Two things not to say
“Pi r squared, pi r squared” with no meaning. Students who chant it use the diameter. Ask them to say which length goes in.
“The area is bigger than the circumference.” For a radius of 2 cm the numbers are equal, and for a smaller radius the area's number is smaller. They are different kinds of quantity, cm² and cm, so neither is bigger (the what-if).
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.
- FluencyA circle has a radius of 10 cm. Find its area. Use 3.14.
314 cm². - FluencyA circle has a radius of 6 cm. Find its area in terms of π.
36π cm².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA garden sprinkler in Lima waters a circle of radius 4 m. How many square metres does it water? Use 3.14.
50.24 m². - SpacingFrom lesson 3: a circle has a diameter of 20 cm. What is its circumference? Use 3.14.
62.8 cm.
3Challenge
Reasoning, generalizing and working backwards.
- MissingA circle has an area of 200.96 cm². Find its radius. Use 3.14.
8 cm: 200.96 ÷ 3.14 = 64, and 8 × 8 = 64. - ReasoningA semicircle has a diameter of 20 cm. Find its area. Use 3.14.
157 cm²: the radius is 10, the whole circle is 314, and half is 157.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Every figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Squared paper, scissors, string and a few sheets of card do the same job: cut, turn, wrap and fold.
Paper circles, scissors and glue are enough for the cut-and-rearrange investigation: 8 sectors gives a wavy strip, 16 a flatter one.