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Circumference of a circle (PDF)

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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

				
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