Area of a circle

Classroom resources

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Students will be able to

• I can explain why the area of a circle is πr².
• I can say why a circle's area is between 2r² and 4r².
• I can find the area from the radius or the diameter.

Timing

Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min

Before class

Do the Run board yourself first. If you can, cut a paper circle into 16 slices and lay them out.

The one idea: the slices make a shape πr long and r high, so A = πr². The common mistake is using 2πr (the circumference) for the area (Jo's mistake).

The board's squares show the bounds 2r² < A < 4r², a quick check on any answer.

Wider: A = ½ × C × r (the rectangle is half the circumference by the radius).

Running the lesson

1. Predict: ask for the outside square's area first.
2. Run: ask what the bumps do as the slices get thinner.
3. Investigate: ask Jo which formula measures round the edge.
4. Modify: ask how many times bigger a radius-10 circle is than radius 5.
5. Make: ask why r² comes before multiplying by π.

The hard part (on purpose)

Keeping area and circumference apart. Ask students to say what each formula measures (a length or a space) before they use it.

When students get stuck

Answers to the checks

1. radius 3: A circle has radius 3 cm. Which is closest to its area?
   28 cm². It is between 2 × 9 = 18 and 4 × 9 = 36 cm²: about 3.14 × 9 = 28.3 cm². (19 cm² is close to the circumference, 18.8 cm, a length.)
2. slices into a rectangle: Rearrange the pieces, then use at least 16 slices. Was your prediction right?
   
   The slices make a near-rectangle πr long and r high: A = πr × r = πr².
3. The long side: Why is the rearranged shape πr long?
   Half the slices' curved edges are along the bottom: half of the circumference 2πr
   The curved edges of the slices make the circumference; half is on top, half on the bottom: πr each.
4. Find the mistake: Jo finds the area of a circle of radius 5 cm. Click the line where her working goes wrong.
   line 2
   Line 2: 2πr is the circumference (a length). The area is πr² = 3.14 × 25 ≈ 78.5 cm².
5. A check with squares: A circle has radius 5 cm. Why must its area be between 50 and 100 cm²?
   It is more than the inside square (2r² = 50) and less than the outside square (4r² = 100)
   The outside square has side 2r (area 4r² = 100); the square on the diagonals has area 2r² = 50.
6. Squares: Compare a circle of radius 4 cm with the squares. Between which two areas is it?
   
   Between 2 × 16 = 32 and 4 × 16 = 64 cm²; πr² ≈ 50.3 cm².
7. Radius 10: Find the area of a circle of radius 10 cm, in cm², to 1 decimal place.
   314.2
   π × 10² = π × 100 ≈ 314.2 cm² (with 3.14: 314 cm²).
8. Double the radius: The radius of a circle doubles. What happens to its area?
   It becomes 4 times as big
   π(2r)² = 4πr²: four times the area.
9. 32 slices: Rearrange 32 slices of a circle. How close is the shape to a rectangle?
   
   With 32 slices the edges are nearly straight: the rectangle is πr by r.
10. Radius 7: Find the area of a circle of radius 7 cm, in cm², to 1 decimal place.
    153.9
    π × 49 ≈ 153.9 cm² (with 3.14: 153.86).
11. From the diameter: A circle has diameter 10 m. Find its area in m², to 1 decimal place.
    78.5
    r = 5 m, so A = π × 25 ≈ 78.5 m².
12. The formula: Which formula gives the area of a circle?
    A = πr²
    A = πr²; the circumference is C = 2πr = πd.
13. Review: circumference: Review: find the circumference of a circle of radius 5 cm, in cm, to 1 decimal place.
    31.4
    C = 2 × π × 5 ≈ 31.4 cm.
14. an area of 50: Find the radius (in steps of 0.5 cm) whose area is closest to 50 cm².
    
    r = 4: π × 16 ≈ 50.3 cm².

Solutions

One possible solution for each task. Other correct programs are fine.

squares and sizes › Squares

Open task: any program that does what the step asks.

squares and sizes › Radius 10

Open task: any program that does what the step asks.

squares and sizes › Double the radius

Open task: any program that does what the step asks.

squares and sizes › 32 slices

Open task: any program that does what the step asks.

an area of 50

Open task: any program that does what the step asks.

Stretch and support

Stretch: Show that A = ½ × C × r using the rearranged slices. Then use it to find the area of a circle with circumference 31.4 cm.

Support: A card with both formulas side by side, labelled length and area.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Area of a circle".

The idea: Deriving the area of a circle formula: I can Students understand why A = πr² using at least one informal derivation method.

What understanding looks like:
- States A = πr² and can explain the formula using at least one informal argument.
- Explains why the area must be between 2r² and 4r².
- Does not confuse area formula with circumference formula.

Mistakes students often make here:
- A student uses 2πr for area.
- A student uses the diameter in πr².
- A student works out (πr)² or π × 2r.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Area of a circle".

Exit ticket

Find the area of a circle with diameter 8 cm.

r = 4 cm; A = π × 16 ≈ 50.3 cm².

What each step is for

• radius 3: Committing to an area for radius 3 shows whether they reason between 2r² and 4r², or mix area with circumference.
• slices into a rectangle: Rearranging more and more slices shows the shape approaching a rectangle πr by r, which gives A = πr².
• why it works: The long side is half the circumference; area and circumference are different formulas (Jo uses 2πr); 2r² < A < 4r² is a check.
• squares and sizes: Comparing with squares, using the formula, and doubling the radius show how area grows with r².
• circle areas: Area from the radius or diameter and a formula choice, mixed; using 2πr for area is the planted slip; the circumference review keeps the other formula separate.
• an area of 50: Searching for a radius to hit an area uses πr² as a prediction before the board checks it.

Where it leads

This starts AC9M8M03.

Where it leads: circle problems (next lesson), cylinders (Year 9) and sectors.