--- title: "Circumference" subtitle: "Area, volume, circles and Pythagoras · Circles · Lesson 8" --- # Circumference **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A bike wheel 70 cm across rolls forward one circumference with every turn. How many turns to school? A trundle wheel measures a sports field the same way. :::teacher-note **In one sentence:** Students use C = πd and C = 2πr to find circumferences, check against circles rolled out on the board, go backwards with d = C ÷ π, and design a trundle wheel (AC9M7M03, node 3). **Students should be able to say:** I can find a circle's circumference from its diameter or radius. I can find a diameter or radius from a circumference. I can give an exact answer in terms of π. **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 Find the diameter first. :::keep-together **A circle has radius 5 cm. About how long is its circumference?** ☐ 10 cm ☐ 15.7 cm ☐ 31.4 cm ☐ 78.5 cm My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 31.4 cm. The diameter is 10 cm, and π × 10 ≈ 31.4. 15.7 uses the radius as the diameter; 10 cm is the diameter itself. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Forwards and backwards :::keep-together ### Use the diameter Here is the formula forwards and backwards. 1. $C = \pi d = 3.14 \times 8 = 25.12 \text{ cm}$ — From the diameter: multiply by π. 2. $C = 2\pi r = 2 \times 3.14 \times 30 = 188.4 \text{ cm}$ — From the radius: double it first (2r is the diameter). 3. $d = C \div \pi = 18.84 \div 3.14 = 6 \text{ m}$ — Backwards: divide by π. 4. $C = 10\pi \text{ cm}$ — Exact: leave π in the answer (for d = 10). 10π means 10 × π, about 31.4. Now try one: diameter 8 cm. **A circle has diameter 8 cm. Using π ≈ 3.14, what is its circumference, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 25.12 3.14 × 8 = 25.12 cm. ::: :::keep-together ### Find the mistake **Mei finds how far a wheel rolls in one turn. Click the line where her working goes wrong.** ``` 1 A wheel has radius 30 cm. How far does it roll in one turn? 2 C = π × d 3 = 3.14 × 30 4 = 94.2 cm ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: 30 cm is the radius, but C = π × d needs the diameter, 60 cm. 3.14 × 60 = 188.4 cm (or C = 2πr = 2 × 3.14 × 30). ::: :::keep-together ### Backwards **A circular garden bed has circumference 18.84 m. Using π ≈ 3.14, find its diameter. Write your steps.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $C = \pi d$, so $18.84 = 3.14 \times d$ (the circumference is π times the diameter); $d = 18.84 \div 3.14 = 6$ m (divide by π to undo multiplying by π) 18.84 ÷ 3.14 = 6 m. ::: :::typical ## One change at a time **Circumference 25.** Each task changes one thing. **With 22/7.** **Exact.** **Backwards: the radius.** **A semicircle.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Circumference :::keep-together ### From the diameter **Diameter 12 cm. Circumference, in cm (π ≈ 3.14)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 37.68 3.14 × 12 = 37.68 cm. ::: :::keep-together ### From the radius **Radius 3 cm. Circumference, in cm (π ≈ 3.14)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 18.84 2 × 3.14 × 3 = 18.84 cm. ::: :::keep-together ### A bike ride **A bike wheel has diameter 70 cm. How far does the bike go in 100 turns, in metres (π ≈ 3.14)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 219.8 One turn: 3.14 × 70 = 219.8 cm. 100 turns: 21 980 cm = 219.8 m. ::: :::keep-together ### Review: area **Review: a triangle has base 10 cm and height 6 cm. Its area, in cm²?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 30 ½ × 10 × 6 = 30 cm². ::: ## Make: a trundle wheel Then work out how many clicks it makes around a 400 m running track. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Find the circumference of a circle with radius 4 cm. A circle has circumference 12.56 cm: find its diameter. (π ≈ 3.14)** {{lines-3}} ::: :::teacher-note **Answer:** 2 × 3.14 × 4 = 25.12 cm. 12.56 ÷ 3.14 = 4 cm. :::