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Mathematics Stage 4 MS4.1 Perimeter and area Part 6 Aout circles State of New South Wales, Department of Education and Training 2004
Contents Part 6 Introduction Part 6...3 Indicators...3 Preliminary quiz Part 6...5 Circumference of a circle...9 Semicircles and quadrants...13 Area of a circle...17 Areas of parts of circles...21 Composite shapes...23 Suggested answers Part 6...27 Exercises Part 6...31 Review quiz Part 6...45 Answers to exercises Part 6...49 Part 6 Aout circles 1
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Introduction Part 6 This is the last part of Perimeter and area and deals with calculating the perimeters and areas of circles, semicircles and quadrants. Here you will use the asic formulas C = 2πr and A = πr 2 and learn how to modify them when dealing with parts of a circle. You will also solve prolems involving π and realise that there are a numer of approximations, such as 3.14 and 22, including the calculator display (giving its value to aout 7 ten decimal places or more if you use the calculator on a computer). Indicators By the end of Part 6, you will have een given the opportunity to work towards aspects of knowledge and skills including: developing, from the definition of π, formulas to calculate the circumference of circles in terms of the radius and diameter: C = 2πr and C = πd developing y dissection and using the formula to calculate the area of circles A = πr 2. By the end of Part 6, you will have een given the opportunity to work mathematically y: finding the area and perimeter of quadrants and semicircles finding radii of circles given their circumference or area solving prolems involving π and an approximate answer using 22 7, 3.14 or a calculator s approximation for π. comparing the perimeter of a regular hexagon inscried in a circle with the circle s circumference to demonstrate that π > 3. Part 6 Aout circles 3
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Preliminary quiz Part 6 Before you start this part, use this preliminary quiz to revise some skills you will need. Activity Preliminary quiz Try these. 1 Use your calculator to write the value of π a c correct to one decimal place correct to two decimal places correct to four decimal places 2 Throughout history various approximations for π have een given. Here are some. Use your calculator to write each of these correct to five decimal places. 22 a 7 = 9.87 = c d 4 3 4 = 355 113 = e 3 8 9 2 = Which of these values is closest to the value of π given y your calculator? Part 6 Aout circles 5
3 Calculate the areas of the following shapes. a A square, side length 7.1 cm. A rectangle, with length 10.5 m, and readth 6.3 m. c A triangle with ase 45 mm, and perpendicular height 27 mm. 4 Use Pythagoras s theorem to calculate the length of the side AB in each of these. (Give answers correct to one decimal place.) a A 17.4 cm C 7.8 cm B C 4.7 m 6.5 m A B What is the perimeter of each of these triangles? First triangle: Second triangle: 6 MS4.1 Perimeter and area
5 Calculate the perimeter and area of this composite shape. Perimeter: Area: 14 cm 20 cm 12 cm 30 cm Check your response y going to the suggested answers section. Part 6 Aout circles 7
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Circumference of a circle In an earlier session you found that the value of circumference diameter circle produces a value a little over three. for a Mathematicians call this numer π (pronounced pie ) and it does not have an exact value. From your calculator you can see that π 3.14159, and for most calculations this is all the accuracy you need. So you can write: circumference = π or using C for diameter circumference, and d for diameter, C d = π. i c r c u m f e r e n c e diameter diameter, d radius, r This gives a useful formula for finding the circumference of a circle: C = πd (circumference = π diameter ). As the diameter is doule the radius, another useful formula is: C = 2πr (circumference = 2 π radius). (Unless you particularly like rememering formulas, just rememer one of these say, C = 2πr, and adjust it when given the diameter.) Part 6 Aout circles 9
Follow through the steps in this example. Do your own working in the margin if you wish. Calculate the circumference of this circle having radius 3.5 cm. 3.5 cm Solution C = 2πr = 2 π 3.5 = 21.99 cm (two decimal places) You do not need to enter 3.14159 for π. Most calculators have a π key. Check with your calculator. You might e ale to access it directly, or you might need to press the SHIFT nd, 2 F, or INV key first. When π is pressed on your calculator the screen fills up with numers. Most of these are not significant. So when you calculated 2 π 3.5 you could have written 21.99114858. But the radius of this circle is 3.5 cm, rounded correct to one decimal place. Rememer, that all measurements are not exact ut approximations to a certain level of accuracy. If the radius is 3.49 cm, the circle s circumference is 21.928 cm (correct to three decimal places). If the radius is 3.51 cm, the circumference is 22.054 cm 3.49 cm c = 21.928 cm 3.51 cm c = 22.054 cm As a simple rule, give your answer to the same numer of decimal places as the radius (or diameter) or, at most, one more decimal place. So the answer to the example aove should e given as 21.99 cm (correct to two decimal places), or 22.0 cm (correct to one decimal place). 10 MS4.1 Perimeter and area
Follow through the steps in this example. Do your own working in the margin if you wish. The circumference of a circle is 50 cm. Calculate its radius. r cm circumference = 50 cm (If a diagram is not provided, draw one and write on all relevant information.) Solution C = 2πr 50 = 2 π r r = 50 2 π = 7.96 cm In this example, the circumference was known ut its radius was not. Write all relevant numers in the formula then rearrange to find r. 50 Be careful when calculating. You need to keep the 2 π together 2 π on your calculator. One way is to use grouping symols: 50 (2 π) =. Always estimate your answer eforehand so you have a fair idea that the answer you arrive at with your calculator appears correct. Developing techniques like this will take a little practice. Mmmm! π = 3, so 50 2 π 50 6 which is aout 8. Part 6 Aout circles 11
Activity Circumference of a circle Try these. 1 Calculate the circumference of the circle given its radius is 12.6 cm. 12.6 cm 2 A circle has circumference 7.5 m. Calculate its diameter. Check your response y going to the suggested answers section. Do some more prolems to reinforce your understanding of the circumference of a circle and its properties. Go to the exercises section and complete Exercise 6.1 Circumference of a circle. 12 MS4.1 Perimeter and area
Semicircles and quadrants The perimeter around the rim of a circle is the circumference and may e calculated using either C = πd or C = 2πr where C is the circumference, d is the diameter and r is the radius. A circle can e divided into two semicircles. The curved part of the semicircle is half the circumference of the whole circle. So the perimeter of a semicircle is: Perimeter = curved part + diameter πr P = 1 2 π r + 2r 2 =πr + 2r Using the diameter instead of the radius, P = 1 2 πd + d diameter = 2r semi-circle The perimeter of a semicircle includes oth the curved portion (which is half the circumference of a complete circle) and the length of the diameter. Follow through the steps in this example. Do your own working in the margin if you wish. Calculate the perimeter of this semicircle, correct to one decimal place. 12.8 cm Part 6 Aout circles 13
Solution The length of the curved part is 1 2 π d = 1 π 12.8 2 = 20.1 cm So P = curved part + diameter = 20.1 + 12.8 = 32.9 cm There is no need to rememer either formula. Just uild up the perimeter y realising you must add together the two portions. Some students prefer formulas using radii. It is easy to change since radius = 1 2 diameter. Activity Semicircles and quadrants Try this. 1 Calculate the perimeter of this semicircle, correct to one decimal place. 7.4 m Check your response y going to the suggested answers section. A circle can e divided into four quadrants. The curved part of the quadrant is one quarter the circumference of the whole circle. So the perimeter of a quadrant is: 14 MS4.1 Perimeter and area
Perimeter = curved part + two radii P = 1 2 π r + 2r 4 = 1 πr + 2r 2 1 2 πr radius, r radius, r Using the diameter instead of the radius, P = 1 4 πd + d quadrant Just as efore, there is no need to rememer these formulas. Just e ale to find the lengths of each part then add them together. Follow through the steps in this example. Do your own working in the margin if you wish. Calculate the perimeter of this quadrant, correct to one decimal place. Solution The length of the curved part is: 1 2 π r = 1 2 π 5.3 = 8.3 m So P = curved part + 2 radius = 8.3 + 2 5.3 = 18.9 m 5.3 m Sometimes a radius will e given, at other times a diameter will e provided. You need to e ale to calculate the perimeter using either. Part 6 Aout circles 15
Activity Semicircles and quadrants Try this. 2 Calculate the perimeter of this quadrant, correct to one decimal place. 21.2 cm Check your response y going to the suggested answers section. Now its time for you to have some more practice solving prolems that involve semicircles and quadrants. Go to the exercises section and complete Exercise 6.2 Semicircles and quadrants. 16 MS4.1 Perimeter and area
Area of a circle In an earlier session you determined that the area of a circle is aout three times the square of the radius. Earlier you also learned that the formula for the circumference of a circle is C = 2πr or C = πd where r and d are the radius and diameter. When you cut up the sectors in = πd or C = the circle and joined them up as shown you produced a shape close to a rectangle. C 2πr This approximate rectangle has a length of half the circumference (πr ) and readth equal to the radius ( r). So its area is length readth: r half circumference = πr readth = r A = πr r length = πr = πr 2 So the area of a circle is: A = πr 2 Follow through the steps in this example. Do your own working in the margin if you wish. Calculate the area of a circle whose radius is 4 cm. (Leave your answer in exact form. That is, with the π in it). r = 4 cm Part 6 Aout circles 17
Solution A = πr 2 = π 4 2 = 16π cm 2 Like all areas, the area of a circle is measured in square units. Here the units are square centimetres. Activity Area of a circle Try this. 1 Calculate the area of the circle, giving your answer in exact form and also correct to one decimal place. 8 cm Check your response y going to the suggested answers section. Even if you were not asked to provide the answer correct to one decimal place, you should limit your answer to at most one or two decimal places, given that the radius was provided to the nearest whole numer. Sometimes you might e given the area of the circle and you need to find its radius (or diameter). 18 MS4.1 Perimeter and area
Follow through the steps in this example. Do your own working in the margin if you wish. Calculate the radius of this circle. r m area = 24.7 m 2 Solution A = πr 2 24.7 = πr 2 r 2 = 24.7 π (dividing oth sides y π) r = 24.7 π = 2.8 m (correct to one decimal place) Sustitute in the values given, then rearrange to find the value of the radius, r. Activity Area of a circle Try this. 2 Calculate the diameter of this circle, correct to two decimal places. (Hint: calculate the radius first.) d cm area = 67.3 cm 2 Check your response y going to the suggested answers section. Part 6 Aout circles 19
Don t round off until the end. Keep the value of the radius to one or two more decimal places. If you round off as you go along you will lose accuracy. Practise these skills more with the prolems in the exercise. Go to the exercises section and complete Exercise 6.3 Area of a circle. 20 MS4.1 Perimeter and area
Areas of parts of circles The area of a circle is A = πr 2. So the area of a semicircle is half this, while the area of a quadrant is one quarter this. A = 1 2 πr2 A = 1 4 πr2 Follow through the steps in this example. Do your own working in the margin if you wish. Calculate the area of this semicircle giving your answer in terms of π, and also correct to one decimal place. 10 cm Solution The radius of the semicircle is 5 cm. A = 1 2 πr2 = 1 2 π 5 2 = 25 2 π cm2 (exact form) = 39.3 cm 2 (one decimal place) Part 6 Aout circles 21
Rememer to check your answer mentally, so you can see if the answer given y your calculator appears correct. As π is aout 3, then: 25 2 π 25 2 3 = 12.5 3 = 37.5 So your answer of 39.3 appears correct. Activity Areas of parts of circles Try these. 1 Calculate these areas, giving your answers in exact form, and also correct to one decimal place. a 24 cm 3 m Check your response y going to the suggested answers section. You should now consolidate this learning y doing the exercise. Go to the exercises section and complete Exercise 6.4 Areas of parts of circles. 22 MS4.1 Perimeter and area
Composite shapes Composite shapes are made up of two or more simple shapes. In earlier parts you dealt with shapes made up of squares, rectangles and triangles. Here you will consider shapes that contain parts of circles. The techniques you learned to find the perimeters and areas of such shapes apply here. Follow through the steps in this example. Do your own working in the margin if you wish. The shaded shape is called an arelos. Calculate the area and perimeter of the arelos, leaving your answers in terms of π. 14 cm 6 cm Part 6 Aout circles 23
Solution To calculate the area There are three semicircles. The largest has a radius of 10 cm 14 + 6 2. Its area is 1 2 π r 2 = 1 2 π 102 = 50π cm 2 The other two semicircles have areas 49 2 π cm2 and 9 2 π cm2. (These circles have radii of 7 cm and 3 cm respectively.) The area of the arelos is the largest area minus the two smaller areas. Area of arelos = 50π 49 2 π 9 2 π = 21π cm2. To calculate the perimeter There are three semicircles, ut you only need to calculate the arc lengths of them. The largest has arc length: 1 2 2 π r = π 10 =10π cm The other two semicircles have arc lengths 7π cm and 3π cm. So the perimeter of the arelos is the sum of the arc lengths. P = 10π + 7π + 3π = 20π cm You need to decide with composite shapes which parts need to e added and which sutracted. By examining the shape first you should e ale to quickly determine this. 24 MS4.1 Perimeter and area
Activity Composite shapes Try this. 1 This shape is made up of a quadrant of a circle and a triangle. a What is the length of BD? B A 15 cm D C Use Pythagoras s theorem to calculate length BC, correct to one decimal place. c Calculate the arc length, AB, of the quadrant, ABD. d Calculate the perimeter of the composite shape. e Calculate the area of triangle BCD. f Calculate the area of the quadrant ABD, correct to one decimal place. Part 6 Aout circles 25
g Calculate the total area of the composite shape. Check your response y going to the suggested answers section. Now you should practise determining the area of composite shapes some more to consolidate your learning. Go to the exercises section and complete Exercise 6.5 Composite shapes Congratulations you have completed the learning for this part. It is now time for you to complete the review quiz so you can show your teacher how much you have learned. 26 MS4.1 Perimeter and area
Suggested answers Part 6 Check your responses to the preliminary quiz and activities against these suggested answers. Your answers should e similar. If your answers are very different or if you do not understand an answer, contact your teacher. Activity Preliminary quiz 1 a 3.1 3.14 c 3.1416 2 a 3.14286 3.14166 c 3.16049 d 3.14159 e 2.37037 the closest value is 355 113. 3 a 50.4 cm 2 66.15 m 2 c 607.5 mm 2 4 a AB = 19.1 cm; perimeter = 44.3 cm AB = 4.5 m; perimeter = 15.7 m 5 Perimeter = 76 cm; area = 264 cm 2 Activity Circumference of a circle 1 C = 2πr = 2 π 12.6 = 79.16813487 = 72.2 cm (one dp) 2 C = 2 π r which can e written as C = π d So d = C π = 7.5 π = 2.4 m (one dec. pl.) Part 6 Aout circles 27
Activity Semicircles and quadrants 1 C = 2 π r which can e written as C = π d So d = C π = 7.5 π = 2.4 m (one dec. pl.) 2 Perimeter = length of curved part + 2 radius = 1 4 2 π r + 2r = 1 π 21.2 + 21.2 2 = 54.5 cm (correct to one decimal place) Activity Area of a circle 1 A = πr 2 = π 8 2 = 64π (exact form) = 201.1 (one dec. pl.) 2 A = πr 2 67.3 = π r 2 r 2 = 67.3 π r = 67.3 π = 4.628 cm Now diameter = 2 radius d = 2 4.628 = 9.26 cm (two dec. pl.) Activity Areas of parts of circles 1 a 72π cm 2, 226.2 cm 2 9π 4 m 2, 7.1 m 2 28 MS4.1 Perimeter and area
Activity Composite shapes 1 a BD = 15 cm (it is the same length as AD) BC 2 = BD 2 + DC 2 =15 2 +15 2 = 450 BC = 450 = 21.2 cm (1 dp) c This is one quarter of a circle. arc length = 1 4 2 π r = 1 4 2 π 15 = 23.6 cm (1 dp) d e P = 21.2 + 23.6 + 15 + 15 = 74.8 cm Area = 1 2 h = 1 15 15 2 =112.5 cm 2 f This is one quarter of a circle. Area = 1 4 π r 2 = 1 4 π 152 =176.7 cm 2 (1 dp) f Total area = 112.5 + 176.7 = 289.2 cm 2 Part 6 Aout circles 29
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Exercises Part 6 Exercises 6.1 to 6.5 Name Teacher Exercise 6.1 Circumference of a circle 1 Calculate the circumference of these circles, correct to one decimal place. a 16.8 cm 8.4 m 2 a Calculate the value of 22 7, to four decimal places. Why is this fraction sometimes used as an approximation for π? Part 6 Aout circles 31
3 Sometimes answers are left in terms of π. These are known as exact answers as the value of π is not approximated to any set numer of decimals places. In the following example the answer is given oth in exact form (in terms of π ), and as an approximation after eing multiplied y a value for π. C = πd = π 15 = 15π (exact answer) = 47.1 (correct to one decimal place) 15 m Calculate these circumferences, leaving answers in exact form (in terms of π ). a 33.1 cm 7.7 m 4 Calculate the diameter and radius of this circle. circumference = 84 cm 32 MS4.1 Perimeter and area
5 A trundle wheel is a simple device for measuring distances that are too long for a tape measure, or that aren t relatively straight. The circumference around the wheel is 1.0 metre. Calculate the radius of the wheel, correct to the nearest millimetre. Part 6 Aout circles 33
Exercise 6.2 Semicircles and quadrants 1 Calculate the perimeter of these semicircles, correct to two decimal places. a 67.3 cm 3.5 m 2 Calculate the perimeter of these quadrants, correct to two decimal places. a 5. 4 cm 123 mm 3 Using π 22 7, calculate the perimeter of this semicircle. 25 cm 34 MS4.1 Perimeter and area
4 How much longer is the path around the circle than the path around the quadrant? (Give your answer correct to one decimal place.) (Hint: calculate the perimeter of each figure first.) 27 m 27 m 5 A regular hexagon is inscried (drawn inside) in a circle. a Why is POQ = 60? P 60 O 10 cm What kind of triangle is POQ? Q c d What is the length of PQ? Calculate the perimeter of the hexagon. e Using C = 2πr, calculate the circumference of the circle. f g By how much is the circumference of the circle longer than the perimeter of the hexagon? If the radius of a circle is r, why is the perimeter of the hexagon 6r? h i Why is 2πr > 6r? _ Explain how this shows that π > 3? Part 6 Aout circles 35
Exercise 6.3 Area of a circle 1 Calculate the areas of these circles, leaving your answer in exact form. a 15 cm 36 m 2 Give the areas of these circles, correct to one decimal place. a _ 5.6 m _ 63 cm 3 Calculate the diameter of a circle with area 100 cm 2. d cm area = 100 cm 2 36 MS4.1 Perimeter and area
4 Calculate the radius of a circle, to the nearest centimetre, having the same area as this square. r cm 1 metre 5 Calculate the area shown shaded, correct to one decimal place. [Hint: first calculate the areas of the square and circle separately.] 6 cm 6 a Calculate the area of this circle using π 3.14159 and π 22 7. (Leave your answer with at least five decimal places.) 2.5 cm _ Comparing your two answers, how many decimal places do you need to go efore the answers are different? Part 6 Aout circles 37
Exercise 6.4 Areas of parts of circles 1 Calculate these areas leaving your answer in terms of π. a 7.6 m 4 m 2 Calculate the areas of these figures giving your answers correct to two decimal places. a 1.9 m 12.7 m 38 MS4.1 Perimeter and area
3 a Calculate the circumferences and areas of these circles, leaving your answers in exact form. Circle 5 cm 10 cm Circumference Area How many times does the circumference increase when the radius of the circle doules? c How many times does the area increase when the radius of the circle doules? Part 6 Aout circles 39
4 Ken and Ester calculated the area of a circle having radius 7 cm. At the us stop they were discussing their answer. As the radius is given to the nearest whole numer my answer is 154 cm 2. My calculator gives π to many decimal places so my answer is 153.93804 cm 2. This is etter than your answer. Is Ester correct? Comment on their answers. 40 MS4.1 Perimeter and area
Exercise 6.5 Composite shapes 1 A circle of radius 7.8 cm is to e cut from a square piece of paper whose side length is 20 cm. a Calculate the area of the square. Calculate the area of the circle, correct to two decimal places. 7.8 cm 20 cm c Calculate the area of the remaining piece. 2 Calculate the area of the composite shapes shaded. a 15 cm 8 cm 12 cm 10 cm 30 cm Part 6 Aout circles 41
3 a Calculate the arc length of the quadrant. 12 cm What is the perimeter of this shape? 4 a Use Pythagoras s theorem to calculate the length PR in PQR. (Give your answer correct to two decimal places.) P Q 3.4 m R Calculate the arc length, PR, of the quadrant correct to two decimal places. c How much longer is arc length, PR, than the straight length, PR? d Calculate the area shaded. 42 MS4.1 Perimeter and area
5 A semicircle lies on the hypotenuse of this triangle. a Calculate the length AB, the diameter of the semicircle. B 10 cm A 24 cm C Calculate the perimeter of this composite shape. c Calculate the area of this composite shape. Part 6 Aout circles 43
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Review quiz Part 6 Name Teacher 1 Calculate the circumference of a circle, leaving your answer in terms of π, having: a radius 11 cm. diameter 7.4 m. 2 The circumference of a circle is 18.5 cm. Calculate its radius, correct to one decimal place. 3 Calculate the perimeter of the semicircle and quadrant, correct to two decimal places. a 54.6 mm 27.5 cm Part 6 Aout circles 45
4 a A circle has a radius of 15 cm. Calculate its area, leaving your answer in exact form. Another circle has a radius of 30 cm. Calculate its area, leaving your answer in exact form. c How many times greater is the area of the second circle compared to the first? 5 A circle needs to e drawn so that its area is exactly 1 m 2. What must its radius e, correct to the nearest millimetre? 6 Calculate the area of the semicircle and quadrant, correct to two decimal places. a 54.6 mm 27.5 cm 46 MS4.1 Perimeter and area
7 a Measure the diameter of a 20-cent coin to the nearest millimetre. Calculate the circumference of this coin. c Descrie one practical method you could use to show that the circumference you calculated is a reasonale measure of the distance around the rim of the coin. 8 Calculate the area and circumference of this composite figure. 6.6 m 15.8 m Area Perimeter Part 6 Aout circles 47
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Answers to exercises Part 6 This section provides answers to questions found in the exercises section. Your answers should e similar to these. If your answers are very different or if you do not understand an answer, contact your teacher. Exercise 6.1 Circumference of a circle 1 a 105.6 cm 26.4 m 2 a 3.1429 π, correct to four decimal places, is 3.1416. So π 22 7, and is equal to two decimal places. (Most times you do not need π to an accuracy greater than this.) 3 a 66.2π cm 7.7π m (Notice how easy it is to leave the answer in terms of π. No calculator involved, and the answer looks neater.) 4 diameter = 26.7 cm, radius = 13.4 cm 5 radius = 15.9 cm (or 159 mm) Exercise 6.2 Semicircles and quadrants 1 a 173.01 cm 18.00 m 2 a 19.28 cm 439.21 mm 3 64.3 cm (correct to one decimal place) 4 Circumference of the circle is longer y 169.64 96.41 = 73.2 m. 5 a POQ is one of six identical triangles in the hexagon. So each angle at the centre is 360 6 = 60. equilateral c 10 cm d 60 cm e 62.83 cm f 2.83 cm (correct to two decimal places) Part 6 Aout circles 49
g h The side length of the hexagon equals the radius of the circle. As the hexagon has six sides, its perimeter is 6 radius = 6r. The circumference of the circle is greater than the perimeter of the hexagon. i 2πr > 6r π > 6r (divide oth sides y 2r) 2r π > 3 Exercise 6.3 Area of a circle 1 a 225π cm 2 324π m 2 2 a 98.5 m 2 3117.2 cm 2 3 11.3 cm (correct to one decimal place) 4 56 cm 5 30.9 cm 2 6 a 19.6349375 cm 2 and 19.64285714 cm 2 two dec. pl. Exercise 6.4 Areas of parts of circles 1 a 7.22π cm 2 4π m 2 2 a 63.34 m 2 2.84 m 2 3 a The 5-cm radius circle: C = 10π cm; A = 25π cm 2 The 10-cm radius circle: C = 20π cm; A = 100 π cm 2 The circumference doules c The area quadruples ( 4 ) 4 Ester is not correct. The radius is a measurement, and all measurements are not exact. It is given to the nearest whole. (It could e 6.9 cm or 7.2 cm, ut was recorded measured to the nearest centimetre.) Ken gives the etter response. Just ecause a calculator provides a value for π to many decimal places does not make the answer any more precise. 50 MS4.1 Perimeter and area
Exercise 6.5 Composite shapes 1 a 400 cm 2 191.13 cm 2 c 208.87 cm 2 2 a 145.1 cm 2 293.4 cm 2 3 a 18.8 cm 54.8 cm 4 a 4.81 cm 5.34 cm c 0.53 cm d 3.30 cm 2 5 a 26 cm 74.8 cm c 385.5 cm 2 Part 6 Aout circles 51