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7 Circumference of a circle This work will help you find circumfere nc • the circumference of a circle from its radius or diameter e • the radius or diameter of a circle from its circumference diameter • the perimeter of a shape involving part of a circle radius You need scissors and a cylindrical object such as a food can. A Finding the circumference of a circle A1 The clock face of Big Ben in London is being cleaned. Use the height of a cleaner to estimate (a) the radius of the clock face (b) the diameter of the clock face (c) the distance the tip of the minute hand moves in five minutes (d) the distance the tip of the minute hand moves in one hour T Make a ‘circumference strip’ for a cylindrical object. Mark the diameter on it. Fold to see how many times the diameter goes into the circumference. What do you find? Is it true for cylindrical objects of any size? 52 7 Circumference of a circle A rough rule is circumference = 3 9 diameter Use this rule to answer the following questions. A2 What is the circumference of each of these, roughly? (a) (b) (c) Shoe Polish p eaches diameter = 10 cm diameter = 7 cm diameter = 6 cm A3 Roughly how much sticky tape is needed to go once round the curved part of each of these parcels? (a) (b) (c) 23 23 cm 15 cm 8 cm A4 What is the circumference of each of these circles, roughly? (a) (b) (c) 2.5 cm 4 cm 5 cm A5 The circles below are not drawn true size, and their radius is given, not their diameter. For each circle, (i) give the diameter (ii) use the rule above to get the rough circumference (a) (b) 6 cm (c) (d) 10 cm 9 cm 8.5 cm 7 Circumference of a circle 53 B Using p When you did the experiment with the ‘circumference strip’ you may have found that the circumference was a little bit more than three times the diameter. For thousands of years, people have tried to work out this ‘3-and-a-bit’ number, always searching for a more accurate value. We now use the Greek letter p (pronounced pie) to stand for this number and we know that it is 3.141 59… A scientific calculator gives p to more decimal places than shown above. Find out how to get p on your calculator (it may involve the SHIFT or second function key). The value on the calculator seems very accurate; 3.14 or 3.142 is accurate enough for most practical purposes. But in fact you can never write p exactly: however many decimal places you write, you will never have the exact value. Do a web search for the word pi. There is a lot about the history of p and how mathematicians have calculated it to more and more decimal places. You’ll also see that it plays an important part in advanced mathematics. But there are also some simple things you can do for fun, like searching for your phone number – or your birth date in figures – in the first two hundred million decimal places of p. Because of the way p fascinates people, you’ll find some eccentric websites. This arrow diagram shows how you work out a circumference from a diameter. diameter 9p circumference A formula connecting C (the circumference) with d (the diameter) is C = pd . We could write C = dp but we usually write numbers (including the number p) before letters when things are multiplied together in algebra. Use the p key on your calculator or 3.142 for the calculations in this section. B1 For each of these circles (i) calculate the circumference and write down the answer your calculator shows (ii) round the answer to one decimal place and show the correct units (a) (b) (c) (d) 2.0 cm 2.6 cm 3.2 cm 3.7 cm 54 7 Circumference of a circle B2 Calculate the circumference of these circles. Write each answer to one decimal place (1 d.p.). (a) (b) (c) (d) 1.8 cm 2.2 cm 3.4 cm 4.4 cm B3 On your calculator, work out the distance round these, rounding your answers to one decimal place. (a) A pipe with diameter 8.0 cm (b) A pipe with diameter 4.2 cm (c) A pipe with diameter 11.2 cm B4 A circular pond has a diameter of 23 metres. Calculate its circumference to the nearest metre. B5 Petra wants to make a bracelet by forming a piece of silver wire into a circle. What length of wire will she need for a bracelet with diameter 8.5 cm? Since the diameter of a circle is twice the radius, this is the arrow diagram for working out the circumference C from the radius r. radius 92 9p circumference A formula connecting C with r is C = 2pr . We could write C = r2p but we usually write numbers (including the number p) before letters when things are multiplied together in algebra. B6 Calculate the circumference of each circle to 1 d.p. (a) (b) (c) (d) 1.4 cm 1.6 cm 0.9 cm 1.9 cm B7 A circular plate has radius 14.0 cm. Calculate its circumference to the nearest 0.1 cm. 7 Circumference of a circle 55 B8 The distance from the centre of a large wind turbine to the tip of a blade is 46 metres. How far does the tip of the blade travel in one rotation? B9 The Earth goes round the Sun once every year, roughly in a circle. The Earth is about 150 million kilometres from the Sun. How far does the Earth travel round the Sun in a year? C Finding a diameter from a circumference If you need to find a diameter from a circumference, the easiest way is to start with the arrow diagram you used earlier. diameter 9p circumference Then reverse it. diameter )p circumference Dividing is the inverse of multiplying. Then use p (or 3.142) and division on your calculator to get the diameter. C1 A circle has circumference of 48.0 cm. Calculate its diameter to 1 d.p. C2 Here are the circumferences of some circles. Calculate the diameter of each circle to 1 d.p. (a) Circumference = 25.2 cm (b) Circumference = 1.5 cm C3 One lap of a circular racing track is 500 m. What is its diameter roughly? C4 Here again is the arrow diagram for finding a circumference from a radius. radius 92 9p circumference (a) Draw a reverse arrow diagram for finding a radius from a circumference. (b) Calculate the radius (to 1 d.p.) of a circle with circumference 14.2 cm. C5 Here are the circumferences of some circles. Calculate the radius of each circle to 1 d.p. (a) Circumference = 34.2 cm (b) Circumference = 2.2 cm C6 A conker is swung in a circle. It travels 240 cm each full turn. How long is the straight part of the string? *C7 The planet Neptune goes round the Sun in roughly a circle. It travels 28 billion kilometres every time it goes round. How far is it from the Sun? 56 7 Circumference of a circle D Checking that a circumference answer makes sense T It helps to be able to check whether answers to circle calculations are sensible. A circle has radius 5 cm. (It is not drawn true size in this diagram.) The sides of the square touch the circle. 5 cm • What is the perimeter of the square? • Which is greater, the perimeter of the square or the circumference of the circle? This regular hexagon is drawn inside the same circle. • What kind of triangle is the shaded triangle? • What is the perimeter of the hexagon? 5 cm • Which is greater, the perimeter of the hexagon or the circumference of the circle? • Calculate the circumference of the circle to 1 d.p. • Does your answer seem sensible compared with the perimeter of the square and the perimeter of the hexagon? D1 A circle has a diameter of 8 cm. (It is not drawn full size here.) (a) Use the appropriate rule to calculate the circumference of the circle, to 1 d.p. 8 cm (b) What is the perimeter of the square? (c) What is the perimeter of the hexagon? (d) Use your answers to (b) and (c) to say whether your answer to (a) is sensible. D2 Find the circumference of each of these circles (they are not drawn accurately). Each time, use the square and hexagon method to check that your answer is sensible. (a) (b) (c) (d) 11 cm 8.4 cm 8 cm 6.8 cm D3 In 1985 a beefburger was made with diameter 2.7 m. What was the circumference of the beefburger? Give your answer to an appropriate degree of accuracy and check that it is sensible. 7 Circumference of a circle 57 E Finding the perimeter of a shape that involves part of a circle When you find the perimeter of part of a circle, or of a shape made from part of a circle and another shape, it is important to do these things. • On a diagram, label key points with letters if they are not already there. • Show all your working, referring to the lettered parts of the shape. In this way you can keep track of your calculations and they can be checked later. Example B Find the perimeter of this quarter circle, to 1 d.p. 2.5 cm The perimeter of this shape has three pieces. So you add three lengths together. A C The circumference of a whole circle with radius 2.5 cm would be 2pr = 2 9 p 9 2.5 = 15.7079… cm So length of arc AC = 15.7079… ) 4 = 3.9269… cm An arc is part of the Length of line segment AB = 2.5 cm circumference of a circle. Length of line segment BC = 2.5 cm So total perimeter = 8.9269… cm = 8.9 cm (to 1 d.p.) E1 Find the perimeter of this semicircle to 1 d.p. P Q 7 cm E2 This shape is three-quarters of a circle. U Find the total perimeter to 1 d.p. 4 cm V W E3 This shape consists of a square with a semicircle attached. F G Find the perimeter of the whole shape to 1 d.p. 6m I 6m H 58 7 Circumference of a circle E4 This shape consists of a quarter circle with a square attached. B C Find the perimeter of the whole shape to 1 d.p. 5 cm A 10 cm D Test yourself T1 Calculate the circumference of this circle. Give your answer correct to one decimal place. 2.8 cm T2 The Barringer crater in Arizona is circular. It has a diameter of 1.6 km. Calculate the circumference of the Barringer crater. OCR T3 A bicycle has a wheel with radius 30 cm. (a) What is the diameter of the wheel? (b) What is the circumference of the wheel? T4 A fence 240 m long is to make a circular pen for sheep. Roughly what will the diameter of the pen be? T5 This shape is made from a semicircle and an equilateral triangle. M Find its perimeter, giving your answer to one decimal place. L 5.6 cm N T6 Silbury Hill in Wiltshire is a prehistoric mound with a circular base. The distance round the base of the mound is 530 m. In 1968 archaeologists dug a tunnel at the base to the centre of the mound. Roughly how long was their tunnel? 7 Circumference of a circle 59