--- title: "Area of a circle" subtitle: "Area, volume, circles and Pythagoras · Circles · Lesson 16" --- # Area of a circle **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** How much pizza is in a pizza? Cut it into slices, lay them out top to tail, and a rectangle appears. :::teacher-note **In one sentence:** Students cut a circle into slices and rearrange them into a near-rectangle πr by r, compare it with squares of r², and use A = πr² (AC9M8M03, node 1). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Imagine the circle inside a square 6 cm by 6 cm. :::keep-together **A circle has radius 3 cm. Which is closest to its area?** ☐ 28 cm² ☐ 19 cm² ☐ 36 cm² ☐ 9 cm² My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 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.) ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Why it works :::keep-together ### The long side **Why is the rearranged shape πr long?** ☐ Because there are n slices ☐ Because π is about 3 ☐ Half the slices' curved edges are along the bottom: half of the circumference 2πr ☐ Because the circle has radius r ::: :::teacher-note **Answer:** 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. ::: :::keep-together ### Find the mistake **Jo finds the area of a circle of radius 5 cm. Click the line where her working goes wrong.** ``` 1 A circle has radius 5 cm. 2 Its area is 2 × π × 5. 3 2 × 3.14 × 5 = 31.4 cm². ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: 2πr is the circumference (a length). The area is πr² = 3.14 × 25 ≈ 78.5 cm². ::: :::keep-together ### 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) ☐ Because 5 × 10 = 50 ☐ Because π is between 2 and 4 exactly ☐ Because the diameter is 10 ::: :::teacher-note **Answer:** 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. ::: :::typical ## Squares and sizes **Squares.** **Radius 10.** **Double the radius.** **32 slices.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Circle areas :::keep-together ### Radius 7 **Find the area of a circle of radius 7 cm, in cm², to 1 decimal place.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 153.9 π × 49 ≈ 153.9 cm² (with 3.14: 153.86). ::: :::keep-together ### From the diameter **A circle has diameter 10 m. Find its area in m², to 1 decimal place.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 78.5 r = 5 m, so A = π × 25 ≈ 78.5 m². ::: :::keep-together ### The formula **Which formula gives the area of a circle?** ☐ A = πr² ☐ A = 2r² ☐ A = πd ☐ A = 2πr ::: :::teacher-note **Answer:** A = πr² A = πr²; the circumference is C = 2πr = πd. ::: :::keep-together ### Review: circumference **Review: find the circumference of a circle of radius 5 cm, in cm, to 1 decimal place.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 31.4 C = 2 × π × 5 ≈ 31.4 cm. ::: ## Make: an area of 50 Then explain why the area is not 3.14 × 4 = 12.56. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Find the area of a circle with diameter 8 cm.** {{lines-3}} ::: :::teacher-note **Answer:** r = 4 cm; A = π × 16 ≈ 50.3 cm². :::