Numeracy. Introduction to Measurement

Transcription

1 Numeracy Introduction to Measurement Te metric system originates back to te 700s in France. It is known as a decimal system because conversions between units are based on powers of ten. Tis is quite different to te Imperial system of units were every conversion as a unique value. A pysical quantity is an attribute or property of a substance tat can be expressed in a matematical equation. A quantity, for example te amount of mass of a substance, is made up of a value and a unit. If a person as a mass of 7kg: te quantity being measured is Mass, te value of te measurement is 7 and te unit of measure is kilograms (kg). Anoter quantity is lengt (distance), for example te lengt of a piece of timber is.5m: te quantity being measured is lengt, te value of te measurement is.5 and te unit of measure is metres(m). A unit of measurement refers to a particular pysical quantity. A metre describes lengt, a kilogram describes mass, a second describes time etc. A unit is defined and adopted by convention, in oter words, everyone agrees tat te unit will be a particular quantity. Historically, te metre was defined by te Frenc Academy of Sciences as te lengt between two marks on a platinum-iridium bar at 0 C, wic was designed to represent one ten-milliont of te distance from te Equator to te Nort Pole troug Paris. In 98, te metre was redefined as te distance travelled by ligt in free space in of a second Te kilogram was originally defined as te mass of litre (know as cubic decimetre at te time) of water at 0 C. Now it is equal to te mass of international prototype kilogram made from platinum-iridium and stored in an environmentally monitored safe located in te basement of te International Bureau of Weigts and Measures building in Sèvres on te outskirts of Paris. Te term SI is from le Système international d'unités, te International System of Units. Tere are 7 SI base quantities. It is te world's most widely used system of measurement, bot in everyday commerce and in science. Te system is nearly universally employed. SI Base Units Base quantity Name Symbol Lengt metre m Mass kilogram kg Time second s Electric current ampere A Termodynamic temperature kelvin K Amount of substance mole mol Luminous intensity candela cd Centre for Teacing and Learning Academic Practice Academic Skills Digital Resources Page ctl@scu.edu.au [last edited on 7 September 07]

2 For oter quantities, units are defined from te SI base units. Examples are given below. SI derived units (selected examples) Quantity Name Symbol Area square metre m olume cubic metre m Speed metre per second m/ s Acceleration metre per second squared m / sec Some SI derived units ave special names wit SI base unit equivalents. SI derived units (selected examples) Quantity Name Symbol SI base unit equivalent Force Newton N m kg / sec Pressure Pascal Pa N / m Work, Energy Joule J Nm Power Watt W J / s Electric Carge Coulomb C As Electric Potential Difference olt W / Celsius (temperature) degree Celsius C K Frequency Hertz Hz /s Capacity litre L (or l) dm If units are named after a person, ten a capital letter is used for te first letter. Often, litres is written wit a capital (L) because a lowercase (l) looks like a one(). An important feature of te metric system is te use of prefixes to express larger and smaller values of a quantity. For example, a large number of grams can be expressed in kilograms, and a fraction of a gram could be expressed in milligrams. Commonly used prefixes are listed in te table below. Multiplication Factor Name Symbol Word form Standard form Power of 0 peta P Quadrillion tera T Trillion giga G Billion mega M Million kilo k Tousand ecto Hundred 00 0 deca da Ten 0 0 deci d Tent centi c Hundredt milli m Tousandt micro µ, mc Milliont nano n Billiont pico p Trilliont A Page

3 Te use of prefixes containing multiples of are te most commonly used prefixes. Using prefixes, conversions between units can be devised. For example: kg = 000g mg = 0.00g Mm = m On te left and side te prefix is used. On te rigt and side te prefix is replaced wit te multiplication factor. On te left and side te prefix is used. On te rigt and side te prefix is replaced wit te multiplication factor. To make te conversion friendlier to use, multiply bot sides by 000 (Wy 000? Because milli means one tousandt and one tousand tousandts make one wole), so 000mg = g. On te left and side te prefix is used. On te rigt and side te prefix is replaced wit te multiplication factor. µ m = m On te left and side te prefix is used. On te rigt and side te prefix is replaced wit te multiplication factor. To make te conversion friendlier to use, multiply bot sides by (Wy ? Because micro means one milliont), so µ m = m Page

4 Numeracy Module contents Introduction Measurement Lengt, Perimeter and Circumference Measurement - Area Measurement - olume Answers to activity questions Outcomes To demonstrate an understanding of te difference between Perimeter, Area and olume. Identify and name plane and solid sapes. Perform calculations involving Perimeter, Area and olume on a variety of sapes. Ceck your skills Tis module covers te following concepts, if you can successfully answer tese questions, you do not need to do tis module. Ceck your answers from te answer section at te end of te module.. (a) Wat units are used to measure perimeter? (b) Wat units are used to measure area? (c) Wat units are used to measure volume?. (a) Find te perimeter of a soccer ground tat is 40 m long and 55 m wide. (b) Find te area of a semicircle wit diameter 5cm. (c) Find te volume in m and L of a square based pyramid wit a base side lengt.4m and eigt.m. (a) Find te perimeter of tis sape (b) Find te area of tis sape 0 cm 4. Te volume of a spere is 00cm, wat is its radius. Centre for Teacing and Learning Academic Practice Academic Skills Digital Resources Page ctl@scu.edu.au [last edited on 7 September 07]

5 Numeracy Topic : Measurement area You migt recall from te units section of tis module tat an area of square metre is a square wit side lengt of metre. Te purpose of tese rules is to provide a simple way of counting te number of square metres, square centimetres or square millimetres witin a sape. For example: 6 cm cm Te rectangle is split up into centimetre divisions. Te rectangle contains 6 x = 8 squares of area cm eac giving an area 8 cm. Tis can be generalised as A= l w Plane Sape Diagram Rule Triangle b b A = b Note: is te perpendicular eigt Square A= s s or s s w Rectangle A = lw l Parallelogram b A = b is te perpendicular eigt (at a rigt angle to te base). Trapezium b a A= ( a+ b ) is te perpendicular eigt or te separation between te parallel sides. Centre for Teacing and Learning Academic Practice Academic Skills Digital Resources Page ctl@scu.edu.au [last edited on 7 September 07]

6 Rombus d d A= dd were d, d are te lengt of te diagonals. Circle r A= π r Were r is te radius of te circle. Example: Find te area of tis parallelogram. 5.5cm 0.cm Te area of a parallelogram is given te expression A = b Te value of b is 0.cm and is 5.5 cm A = b A = A = 56.cm Example: Find te area of tis circle. 9 m Te area of a circle is given te expression A= π r Te value of r is 9 m A= π r A = π 9 A = m ( to d. p.) Example: Find te area of tis trapezium. Te measurements given are in different units, so a conversion is necessary. Tis is a square trapezium so te separation between te parallel sides is equal to one of te sides. (Te 6 mm side) 6 mm.5 cm 5 cm Te area of a trapezium is given by te expression A= ( a+ b ) Te value of is 6 mm. Te lengts of te parallel sides are.5 cm and 5 cm, converted to mm are 5 mm and 50 mm. Page

7 Example: Find te area of tis composite sape. A= ( a+ b ) A = (5 + 50) 6 A = 680 mm or 6.8cm using te conversion 00mm = cm Te composite sape is a semicircle plus a rectangle. Te dimensions of te rectangle are 0 cm and 4 cm. Because te diameter is te same as te lengt of te rectangle, te radius is 4 cm = 7 cm. 4 cm 0 cm Total Area = π r + lw = π + = = 47.96cm Example: Find te radius of a circle tat as an area of 40m? r A= π r 40 = π r 40 π r = π π 40 = r π 40 = r π 40 = r π r.568 Example: If fertiliser is applied at a paddock at a rate of 500kg/a and te paddock is 00m long and 75m wide, wat is te amount of fertilizer to be bougt? Te area of te paddock is: A= l w A = A= 5500m Te area units must be in a, as tere are 0 000m in a, conversion to a is performed by dividing by 0 000, A = A = 5.5a Te rate 500kg/a means 500kg for a, so te amount required is 5.5a x 500kg/a = 65 kg or.65 t. ideo Area Page

8 Activity. Draw a sketc diagram for eac and calculate te area. (a) A square wit a side lengt of 0 cm. (b) A rectangle wit lengt of 0.6m and widt 0cm (c) An rigt angled triangle wit side lengts 7 cm, 5 cm and 44. cm (d) A circle wit a radius of 45mm (e) An isosceles triangle wit sides 47mm and 75mm (x) (f) A semicircle wit a diameter.m. Calculate te area of tese sapes. (a) 9 m (b) m 4.5 m cm 4 m (c) 5cm (d) 95 cm A parallelogram cm 5 cm (e) 5 m.5 m (f) 6 cm 6 cm Page 4

9 (g) 8.5m 8.5m 8.5m 8.5m () 9 m 6 m 8 m Page 5

10 Numeracy Topic : Measurement lengt, perimeter and circumference Naming Plane Sapes (polygons) and teir properties Triangles ( sides) Equilateral equal sides, equal angles (all 60 ) Isosceles equal sides, equal angles. Scalene no equal sides Rigt angled contains a rigt angle Quadrilaterals (4 sides) Square all sides equal in lengt, all angles 90, diagonals equal in lengt and cut at 90 Rectangle opposite sides equal, all angles 90, diagonals equal in lengt Parallelogram opposite sides equal in lengt and parallel, opposite angles equal Rombus all sides equal in lengt and parallel, opposite angles equal, diagonals cut at 90 Trapezium one pair of parallel sides Kite two pairs of adjacent sides equal in lengt, one pair of opposite angles equal, diagonals cut at 90 T l t (5 id ) (6 id ) t (7 id ) Te perimeter of a sape is te distance around te outside of tat sape. Te sape may consist of straigt lines or curves. Te perimeter of a circle is given te special name of circumference. Centre for Teacing and Learning Academic Practice Academic Skills Digital Resources Page ctl@scu.edu.au [last edited on 7 September 07]

11 Plane Sape Diagram Perimeter Any polygon Triangle s s s s s 4 6 s 5 s s s General sum of side lengts P= s+ s + s+ s s n For a scalene triangle P= s+ s + s For an isosceles P= s+ s For an equilateral triangle P= s Square P= 4 s -were s is te side lengt. s Rectangle w P= l+ w or P= ( l+ w), -were l is te lengt and w is te widt. l Parallelogram l s P= l+ s- were l is te lengt of te base and s is te sloping eigt Circle c r C = π r- were r is te radius of te circle. Lengt, perimeter and circumference are all measures of lengt so te units for answer will be lengt units suc as millimetres, centimetres, metres, kilometres Example: Find te perimeter of tis regular exagon. m A regular exagon as 6 equal lengt sides (sown by te dases), so te rule can be written down as; P= 6s P = 6 P= 7m If units are not te same, conversions must take place before calculating te perimeter. Answers can be expressed eiter unit. Example: Find te perimeter of tis scalene triangle. Page

12 45mm.5cm 8.5cm 8.5cm = 85mm,.5cm = 5mm, in mm, te side lengts are 45,85 and 5. P= s+ s + s P = P = 445 mm or 44.5cm Wen doing measurement questions, te steps to a solution involve Drawing a diagram Writing a rule or formula (expression) Substituting in value Obtaining an answer including units Example: Find te circumference of te circle below. 8.cm Te rule for te circumference of circles is: C = π r Rule C = π 8. Substitution C = 76.6 cm (to d.p.) Answer wit units Some sapes are a fraction of a basic sape. Example: Find te perimeter of te semi-circle below. 0 cm Te perimeter of tis sape is made up of parts: semi-circumference and te diameter. Te radius of tis circle is a alf of 0 cm = 0 cm P= π r+ d P P = π = 5.4 cm (to d.p.) Some sapes are composite sapes, made up of two or more basic sapes. Altoug not all measurements are given, tey can be calculated from te measurements given. Example: Find te perimeter of te sape below. cm Tis sape as 0 sides, 8 sides measure cm. Te top side measurement is equal to four cm lengts and te rigt side is equal to two cm lengts. P= 8s+ s + s P = P = P = 4cm Page

13 Sometimes te perimeter is given and a side lengt as to be calculated. Example: Find te side lengt of a square tat as a perimeter of 0 m. s P= 4 s 0 = 4 s s = = s Te side lengt is 0 m Example: Find te radius of te semicircles on te 400 m running circuit drawn below. 80 m r Te perimeter is made up straigts and semicircles lengts (= full circle circumference) Perimeter = 80 + π r 400 = 60 + π r = π r 40 = π r 40 π r = π π 8. = r Te radius of te semicircles is approx 8. m ideo Perimeter Page 4

14 Activity. Draw a sketc diagram for eac and calculate te perimeter. (a) A square wit a side lengt of 0 cm. (b) A rectangle wit lengt of 0.6m and widt 0cm (c) An equilateral triangle wit side lengt 7cm (d) A circle wit a radius of 45mm (e) An isosceles triangle wit sides 47mm and 75mm (x) (f) A semicircle wit a diameter.m. Calculate te perimeter of tese sapes. (a) 9 m (b) m 4.5 m cm (c) (d) 4 m 95 cm cm A parallelogram cm 5 cm (e) 5 m 4 m (f) 6 cm 6 cm Page 5

15 Numeracy Topic : Measurement volume You migt recall from te units section of tis module tat a volume of cubic metre is a cube wit side lengt of metre. Te purpose of tese rules is to provide a simple way of counting te number of cubic metres, cubic centimetres or cubic millimetres witin a sape. For example: 6 cm cm cm Te rectangular prism is split up into centimetre divisions. Te rectangular prism contains 6 x x = 6 cubes of volume cm eac giving a volume of 6 cm. Tis can be generalised as = l w Te first group of solids being considered are te prisms. A prism is a dimensional sape suc as a rectangle, triangle, circle. tat is extended into te tird dimension. Tis solid will ave te same cross sectional area along its eigt. Cross sectional area eigt Tis is a general prism. Te cross sectional area is saded and te eigt is sown. Te volume of any prism is = A, were A is te cross prism sectional area and is te eigt of te prism. Te cross sectional area of te prism can be one of te plane (flat) sapes used above. Cross sectional area eigt Te solid drawn is a rectangular prism because te cross sectional area is a rectangle. Using te volume of a prism expression prism = A and wit te area of a prism A= l w, te volume of a rectangular prism can be written as prism = A = l w Centre for Teacing and Learning Academic Practice Academic Skills Digital Resources Page ctl@scu.edu.au [last edited on 7 September 07]

16 Prism Diagram Rule Triangular Prism Rectangular Prism Cube (prism) w b Cross sectional area l = b prism Note: tere are two values: te first is te perpendicular eigt of te triangle on te end and te second is te eigt of te prism = A = l w s = s s s = s s s Cylinder (circular prism) r = A = π r Example: Find te volume in cm and ml of te can drawn below..8cm Fruity Soft Drink cm Te volume of a cylinder is given by te expression = π r r =.8 cm = cm = π.8 = cm or 544.4mL Example: Find te volume of te sape below. Te area of te end of te prism is a composite sape consisting of a triangle and a rectangle. Page

17 .75m.5 m.4 m 5.7 m Te area of te end is: A = ( l w) + b A = (.5.4) A = A=.465m Te volume is: = A = = or 65.4 to d.p. m Example: Find te volume of water required to fill tis pool. Note tat te pool as a sallow and deep end adding to te complexity of te question. 0m Sallow End 0.9m 5m Deep End.4m Not to Scale Te saded end of te pool is a trapezium, wit te parallel sides being 0.9m and.4m. Te side are separated by 5m. A= ( a+ b ) A = ( ) 5 A= 4.5m Te volume is = A = = 4.5 m or 4.5 klitres or litres ( Remember m ) = kl Example: Find te volume of concrete required to construct tis pipe. 5 cm 0 cm.8 m Finding te area of te annulus: A = A A end large circle small circle A= πr πr A = π 0 π 5 A = A = 549.7cm Tis end sape is called an annulus. To find te volume, te lengt must be converted to cm, wic is 80 cm. Page

18 = A = = 98946cm As pre-mix concrete is usually ordered in m, a conversion is required. Tere are 00 cm = m, tat is cm = m, dividing by is required. olume required is m or approximately 0.m Pyramids A pyramid is a solid, tree dimensional sape wit a plane sape as its base and te sides coming to a single vertex at te top. Tis is a exagonal pyramid because te base is a exagon and te sides come up to a single vertex. Tis is a rectangular pyramid. Te volume of any pyramid is Te volume of a square pyramid is Te volume of a rectangular pyramid is Te volume of a cone is = π A = s = lw = were A is te area of te base. r Example: Find te volume of te cone below..8 cm 5. cm Area of base is: A= π r A = π 5. A = 84.95cm olume of te cone is : = A = = 90.77cm Page 4

19 Spere A spere is a ball like figure. Te volume of a spere is = 4 π r r Te radius of te Eart is 678 km, find te volume of te Eart assuming it is perfectly sperical. 678 km 4 = π r 4 = π 678 =.09 0 km or km ideo olume ideo Oter Ideas Page 5

20 Activity. Draw a sketc diagram for eac and calculate te volume. (a) A cube wit a side lengt of 0 cm. (b) A rectangle prism wit lengt of 0.6m, widt of 0cm and eigt 500mm. (c) A prism wit a base area of 8 m and a eigt of.7m. (d) A cylinder wit a diameter of 50 cm and a eigt of 60 cm. Give your answer in ml and L. (e) A prism wit a semicircular base ( radius = 5 cm) and a eigt of 6cm. Calculate te volume of tese solids. (a) Express te answer in 65 mm.5 m i mm ii cm iii m 55 mm (b).5 m Tis is a water tank. Express your answer in L..8 m (c) m Hint: olume of large cone subtract te volume of small cone. 4 m m (e) A farmer as a large dam covering a of land. Te average dept wen full is m. Calculate te volume of water in dam in kl and ML. Page 6

21 (f) 500 mm. m A farmer as built tese trougs to old water for livestock. How many litres of water will eac troug old? (g) 0 mm 90 mm Tis is an ice-cream cone. Calculate te volume of soft serve ice cream tat would fill te cone and a semisperical amount on top of te cone. () A garden ose as an internal diameter of 5 mm and a lengt of 5 m. Wat volume of water fills te ose? (i) A can contains 75 ml of soft drink and must ave a eigt of 0 mm. Wat will te diameter need to be? Page 7

22 Numeracy Answers to activity questions Ceck your skills. (a) Lengt units suc as mm, cm, m, km (b) (c) Area units suc as olume units suc as mm, cm, m, a or km mm, cm, m, km. (a) P= l+ w P = P= 90m (b) A= π r 7.5 A = π A= 88.6m (c) = s.4 =. =.7m or.7kl or 7L. (a) Te perimeter is basically te circumference of two circles wit a diameter of 0cm. If te diameter is 0 cm, te radius is 5 cm. Te circumference C = x.4 x 5 =.4cm Te perimeter of te sape is x.4 = 6.84 cm (b) Te area of te sape is circles wit a radius of 5 cm and a square wit side lengt 5 cm. Area of circle =.4 x 5 = cm Area of square = 0 x 0 = 00 cm Total area = x = cm Centre for Teacing and Learning Academic Practice Academic Skills Digital Resources Page ctl@scu.edu.au [last edited on 7 September 07]

23 4. 4 = π r 4 00 = π r 00 = r 4 π 00 = r 4 π r =.6cm A spere wit radius.6cm will ave a volume of 00cm. Measurement - lengt, perimeter and circumference. Draw a sketc diagram for eac and calculate te perimeter. (a) A square wit a side lengt of 0 cm. P= 4s P = 4 0 P = 0cm (b) A rectangle wit lengt of 0.6m and widt 0cm (0.6m = 60cm) (c) An equilateral triangle wit side lengt 7cm P= ( l+ w) P = (60 + 0) P = 80 P = 60cm P= s P = 7 P = 8cm (d) A circle wit a radius of 45mm C = π r C = π 45 C = 8.7mm (e) An isosceles triangle wit sides 47mm and 75mm (x) P= s + s P = P = 97mm (f) A semicircle wit a diameter.m P= π r + d P = π P=.085m. Page

24 (a) Unknown sides are 9m 4m= 5m 9 m and 4.5m m=.5m. m 4.5 m P = P= 7m 4 m (b) 60 P= π r+ r P = π cm 6 P = 5.6cm (c) P= l+ s cm P = cm P = 56 (d) Missing side is 5+ (radius) = 57 P= π r cm 4 P = 7.7cm 5 cm (e) Radius =m 4 m 5 m P= π r+ s P = π + 5 P= 6.8m (f) 6 cm 6 cm Perimeter is equivalent to te circumference of a small circle (r=) plus alf te circumference of te big circle (r=6) P= πr+ πr P = π + π 6 P = 7.7cm Area. Draw a sketc diagram for eac and calculate te area. Page

25 (a) (b) A square wit a side lengt of 0 cm. A rectangle wit lengt of 0.6m and widt 0cm A= s s A = 0 0 A = 900cm A= l w A = 60 0 A = 00cm (c) An rigt angled triangle wit side lengts 7 cm, 5 cm and 44. cm Te base and eigt are 7cm and 5cm, te 44.cm side is unnecessary. A= b A = 7 5 A = 47.5cm (d) A circle wit a radius of 45mm A= π r A = π 45 A = 66.7mm or 6.67cm (e) An isosceles triangle wit base 47mm and eigt 7.mm A= b A = A = 67.mm = 6.7cm (f) A semicircle wit a diameter.m (if d=.m, ten r=0.6m) A= π r A = π 0.6 A= 0.565m A = 5650cm. Calculate te area of tese sapes. (a) 9 m m 4.5 m A = l w+ l w A A T T T = (9 4) = 8m 4 m Page 4

26 (b) (c) cm 0 5cm 5cm A = A = π 5 6 A =.cm A= b 6 A = 95 5 A = 75cm π r A parallelogram (d) AT = π r + l w 4 cm AT = π cm A = 94.4cm T (e).5 m 5 m A = A A saded triangle semicircle Asaded = b π r Asaded = 5 5 π.75 A= 07.7m (f) (g) 8.5m 6 cm 8.5m 6 cm 8.5m Te area of te small semicircle is te same as te area of te semicircle missing from te large semicircle, only te area of te large semicircle is required. A= π r A = π 6 A = 56.55cm Te saded area is one quarter of te difference is area between te area of te square and te circle. A= ( s π r ) 4 A = A=.876m ( π ) 8.5m Page 5

27 () 6 m 8 m 9 m Te saded area is te area of te trapezium in te centre add te area of te large semi-circle subtract te area of te small semicircle. Te trapezium as parallel side lengts 6m and 9m separated by a lengt 8m. A= ( a+ b) + πr πr A = ( 6 + 9) 8 + π 4.5 π A = A= 77.66m olume. Draw a sketc diagram for eac and calculate te volume. (a) A cube wit a side lengt of 0 cm. = s s s = = 8000cm (b) A rectangle prism wit lengt of 0.6m, widt of 0cm and eigt 500mm. (0.6m = 60cm, 500mm = 50cm) = l w = = cm (c) A prism wit a base area of 8 m and a eigt of.7m. = A = 8.7 = 75.6m (d) A cylinder wit a diameter of 50 cm and a eigt of 60 cm. Give your answer in ml and L. = A = π r = π 5 60 = 780cm = 780mL = 7.8L (e) A prism wit a semicircular base ( radius = 5 cm) and a eigt of 6cm. Calculate te volume of tese solids. = A = π r = π = cm 5 6 Page 6

28 (a) Express te answer in (i) mm (ii) cm (iii) m = A = btriangle prism 65 mm.5 m = = cm 55 mm = = mm = = m (b).5 m.8 m Tis is a water tank. Express your answer in L. ( r =.5 =.5 m) = A = π r = π = 8.98m.5.8 = 8.98kL = 8 98L (c) m m 4 m olume of large cone (=m, d=m) subtract te volume of small cone (=8m, d=m). = πrh πr = π.5 π 8 = = 9.89m (e) A farmer as a large dam covering a of land. Te average dept wen full is m. Calculate te volume of water in dam in kl and ML. ( a = 0 000m ) (f) 500 mm = A = 0 000m m = m = kL = 90ML A farmer as built tese trougs to old water for livestock. How many litres of water will eac troug old?. m Page 7

29 (g) 0 mm 90 mm = π r = π = 0.8m 0.5. = 0.8kL = 8L Tis is an ice-cream cone. Calculate te volume of soft serve ice cream tat would fill te cone and a semi-sperical amount on top of te cone. Using cm, = + T cone emispere 4 T = πr+ πr T = π 9+ π = T T = 4.cm () A garden ose as an internal diameter of 5 mm and a lengt of 5 m. Wat volume of water fills te ose? Te ose is a cylinder r=7.5mm=0.75cm, =500cm = π r = π = 88.6cm = 88.6cm = 88.6mL (i) A can contains 75 ml of soft drink and must ave a eigt of 0 mm. Wat will te diameter need to be? (75 ml=75cm ) (0mm=cm) = π = r π r = r π r = π 75 r = π r =.54cm d = 6.cm Page 8

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