--- title: "Circle problems" subtitle: "Area, volume, circles and Pythagoras · Circles · Lesson 17" --- # Circle problems **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A running track, a round garden bed, a pizza deal: is a 30 cm pizza more than twice a 20 cm one? Circle problems need the right formula, sometimes run backwards. :::teacher-note **In one sentence:** Students choose between circumference and area, find a radius from an area, and solve problems with semicircles, quarter circles, rings and tracks (AC9M8M03, node 2). **Students should be able to say:** I can choose the circumference or the area formula for a problem. I can find the radius from an area. I can solve problems with parts of circles. **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 What times itself, times π, makes 12.57? :::keep-together **A round rug has an area of 12.57 m². What is its radius?** ☐ 2 m ☐ 3.14 m ☐ 4 m ☐ 6.28 m My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 2 m. 12.57 ÷ π = 4 = r², so r = √4 = 2 m. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Backwards and in parts :::keep-together ### Backwards Here is the routine. 1. $78.5 \div \pi \approx 25$ — Undo the π: this is r². 2. $r = \sqrt{25} = 5\ \text{cm}$ — Undo the square with a square root. 3. $\pi \times 5^2 \approx 78.5$ — Check by going forwards. **A circle has an area of 113.1 cm². Find its radius in cm.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6 113.1 ÷ π ≈ 36, and √36 = 6 cm. ::: :::keep-together ### Find the mistake **Leo finds the radius of a circle with area 50 cm². Click the line where his working goes wrong.** ``` 1 A circle has an area of 50 cm². 2 A = πr², so r = 50 ÷ 3.14. 3 r = 15.9 cm. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: 50 ÷ 3.14 is r², not r. Take the square root: r = √15.9 ≈ 4.0 cm. ::: :::keep-together ### Semicircles **How do you find the area of a semicircle of radius 4 cm?** ☐ Half of π × 4² ☐ π × 4² × 2 ☐ Half of 2 × π × 4 ☐ π × 2² ::: :::teacher-note **Answer:** Half of π × 4² A semicircle is half the circle: ½ × π × 16 ≈ 25.1 cm². ::: :::typical ## Tracks and rings **A semicircle.** **A track's area.** **A track's perimeter.** **Over 100.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Circle problems :::keep-together ### Back to the radius **A circular pool cover has an area of 154 m². Find its radius in m.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 154 ÷ π ≈ 49, and √49 = 7 m. ::: :::keep-together ### A quarter circle **Find the area of a quarter circle of radius 10 cm, to 1 decimal place.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 78.5 ¼ × π × 100 ≈ 78.5 cm². ::: :::keep-together ### A ring **A ring is between circles of radius 5 cm and 3 cm. Find its area in cm², to 1 decimal place.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 50.3 π × 25 − π × 9 = π × 16 ≈ 50.3 cm². ::: :::keep-together ### A semicircle's perimeter **Find the perimeter of a semicircle of radius 5 cm (the curve and the diameter), to 1 decimal place.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 25.7 Half the circumference, π × 5 ≈ 15.7 cm, plus the diameter 10 cm: 25.7 cm. ::: ## Make: a garden bed Then work out how much edging the bed needs (its circumference), and say which value of π you used. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A circular pond has an area of 28.3 m². What is its radius?** {{lines-3}} ::: :::teacher-note **Answer:** 28.3 ÷ π ≈ 9, and √9 = 3 m. :::