VOLUMES. The volume of a cylinder is determined by multiplying the cross sectional area by the height. r h V. a) 10 mm 25 mm.

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1 OLUME OF A CYLINDER OLUMES Te volume of a cylinder is determined by multiplying te cross sectional area by te eigt. r Were: = volume r = radius = eigt Exercise 1 Complete te table ( =.14) r a) 10 mm 5 mm b) 0 cm 1 mm c) 5 m 6.84 m d) 1 mm 45.5 cm Now ceck your answers. Page 1 of 15

2 OLUME OF A CONE Te volume of a cone is 1 te volume of a cylinder into wic te cone would fit exactly. 1 r Were: = volume r = radius = eigt (perpendicular) Note tat te eigt is measured perpendicularly (at rigt angles) to te base. Exercise Complete te table ( =.14) r a) 0 mm 50 mm b) 10 cm 0.5 m c) 5 m m d) 5 mm cm Now ceck your answers. Page of 15

3 OLUME OF A SPHERE 4 r Were: r = radius r Exercise Complete te table ( =.14) r a) 5mm b) 4 m c) 1500 mm Now ceck your answers. Exercise 4 Calculate te volume of te plumb-bob sown above. All dimensions are millimetres. Now ceck your answers. Page of 15

4 Exercise 5 Calculate te volume of te dowel sown above. All dimensions are in millimetres. (Note: SR 7.5 = sperical radius 7.5) Now ceck your answers. Exercise 6 Calculate te volume of te cast iron roller. Linear dimensions are in millimetres = angular dimensions are in radians. Now ceck your answers. Page 4 of 15

5 SUMMARY a) Area of a triangle = 1 (base x eigt) (Note: te eigt is measured at rigt angles to te base). b) Area of a sector = r 60 Wen te angle of is measured in degrees. 1 c) Area of a sector = r Wen te angle of is measured in radians. d) olume of a cylinder = r 1 e) olume of a cone = r (Note: te eigt is measured at rigt-angles to te base). f) olume of a spere = 4 r Please Note: i) In all te above r = radius and = eigt (or lengt if te figure lies orizontally). ii) You must not mix dimensional units in any of te above formula. For example you must not work te radius in millimetres and te eigt in centimetres. Bot radius and eigt must be in eiter millimetres or in centimetres. Page 5 of 15

6 ANSWERS Exercise 1 r a) 10 mm 5 mm 7855 mm b) 0 cm 1 mm cm mm c) m 5 m 6.84 m d) 1 mm Te Answers are in bold. a) r =.14 x 10 x 5 = 7855mm 10 cm 100 mm 45.5 cm b) Te radius is in cm and te eigt is in millimetres. You must not mix tem wen substituting in te formula. i) Working in cm r =.14 x 0 x 1. (1mm = 1.cm) = cm ii) Working in mm r =.14 x 00 x 1 (0cm = 00mm) = mm Page 6 of 15

7 c) Te formula as to be transposed to find r. So r r r r = 4 = m d) Tere is bot transposition and mixed units at te same time. r r i) Working in cm 45.5 cm 1. x (1mm = 1.cm) ii) Working in mm mm 1.14 (1 cm = 1000 mm ) Now return to te text. Page 7 of 15

8 Exercise r a) 0 mm 50 mm mm b) 10 cm 0.5 m cm m c).51 m 5 m m Te Answers are in bold. d) 5 mm 10 cm 100 mm cm 1 a) r = = mm b) i) Working in cm 1 r = (0.5m = 50 cm) = cm ii) Working in m 1 r = (10cm = 0.1m) = m Page 8 of 15

9 c) Te formula as to be transposed. 1 r r = x.14x5 = 6. 0 =.51mm d) Tere is bot transposition and mixed units at te same time. 1 r r i) Working in cm = (5mm =.5 cm) = 10 cm ii) Working in mm = ( 1 cm = 1000 mm ) Now return to te text = 100mm Page 9 of 15

10 Exercise r a) 5mm mm b) m 4 m c) 7.1 mm 1500 mm Te Answers are in bold. a) 4 r = 4 x.14 x 5 = mm b) Tis time you ave to transpose te formula. 4 r r 4 r 4 r x4 4x.14 = = m c) Transpose te formula. r 4 x1500 r 4x.14 Now return to te text. Page 10 of 15

11 Exercise 4 Te plumb-bob is made up from two geometrical sapes: A cylinder A cone Cylinder r Cone =.14 x 15 x 40 Note: te diameter is 0mm So te radius is 15mm = mm 1 r = 1 x.14 x 15 x (75-40) = 1 x.14 x 15 x 5 = 847.8mm olume of plumb-bob = olume of te cylinder plus olume of te cone olume of plumb-bob = Now return to te text = mm Page 11 of 15

12 Exercise 5 Te dowel consists of tree geometrical sapes. A cylinder A emispere ( ½ a spere) A frustum of a cone ( a cone wit te top cut off) Recognising te sape is alf te battle. Cylinder r =.14 x 7.5 x 60 = mm Hemispere 1 4 r = 1 x 4 x.14 x 7.5 Frustum = 88.69mm Te volume of te frustum is te difference between te volumes of te two cones. We also ave to use some trigonometry to determine te dimensions of te cones. olume of frustum = volume cone A volume cone B Page 1 of 15

13 Since te included angle of te nose of te cone is 60 0, and it is symmetrical, a slice troug te cone on its centre line is an equilateral triangle. All sides 15mm, all angles Tere are various ways of finding using trigonometry or Pytagoras you pays your money and takes your pick. Let s practice our trigonometry. = 15 cos 0 = 15 x = 1.99 mm olume (cone A) 1 r = 1 x.14 x 7.5 x 1.99 = mm Before we can find te volume of cone B, we ave to find its base radius r. r = (-4) tan 0 = (1.99-4) x = 5.19 mm Page 1 of 15

14 olume (cone B) = 1 r = 1 x.14 x 5.19 x 8.99 = 5.6 mm Terefore te volume of te frustum = mm 5.6 mm = mm olume of cylinder = mm olume of emispere = mm olume of frustum = mm Te total volume of te dowel = mm For all practical purposes 1000 mm Te most likely pit falls are: Using te diameter (15mm) instead of te radius (7.5mm); Failing to plan your operation sequence so tat eac step produces te data needed in te next step; Not recognising te basic geometrical figures wic combine togeter to make te dowel. Now return to te text Page 14 of 15

15 Exercise 6 To answer tis you ave to find te volume of te wole roller and ten subtract te volume of te centre ole and te ligtening oles. Roller blank olume = r (diameter = 80 mm, so radius = 40 mm) Centre ole =.14 x 40 x 0 = mm olume = r (diameter = 15mm, so radius = 7.5 mm) =.14 x 7.5 x 0 = mm Ligtening ole To find te volume of one of te ligtening oles multiply te profile area by te tickness (0mm). Te profile area is te difference between two sectors. Profile area = ( 1 R 0) ( 1 R 0) = ( 1 x 0 x 1.0) ( 1 x 0 x 1.0) = = 50 mm olume of ligtening ole = 50 mm x 0 mm = 5000 mm Total volume of te four ligtening oles = 5000 x 4 = 000 mm Terefore, te volume of te roller is: mm mm 000 mm olume = mm Now return to te text. Page 15 of 15