Mensuration - III (Surface Area and Volume of a Right Circular Cylinder) | Mathematics (Maths) Class 8 PDF Download

 Page 1


Q u e s t i o n : 1
Find the curved surface area and total surface area of a cylinder, the diameter of whose base is 7 cm and height is 60 cm.
S o l u t i o n :
Let r and h be the radius and the height of the cylinder. Given: r =
7
2
 cmh = 60 cmCurved surface area of the cylinder = 2 p ×r×h                                               = 2 ×
22
7
×
7
2
×60                          
Q u e s t i o n : 2
The curved surface area of a cylindrical road is 132 cm
2
. Find its length if the radius is 0.35 cm.
S o l u t i o n :
Consider h to be the height of the cylindrical rod. Given: Radius, r = 0. 35 cmCurved surface area = 132 cm
2
We know: Curved surface area = 2 × p ×r×h                       132 = 2 ×
22
7
×0. 35 ×h 
Q u e s t i o n : 3
The area of the base of a right circular cylinder is 616 cm
2
 and its height is 2.5 cm. Find the curved surface area of the cylinder.
S o l u t i o n :
Given: Area of the base of a right circular cylinder = 616 cm
2
Height = 2 .5 cm Let r be the radius of the base of a right circular cylinder. pr
2
 = 616 ? r
2
= 616 ×
7
22
? r
2
= 196 ? r = 14 cm
Q u e s t i o n : 4
The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find its curved surface area and total surface area.
S o l u t i o n :
Given: Height, h = 15 cmCircumference of the base of the cylinder = 88 cm
2
Let r be the radius of the cylinder. The circumference of the base of the cylinder = 2 pr88 = 2 ×
22
7
×rr =
88×7
2×22
= 14 
Q u e s t i o n : 5
A rectangular strip 25 cm × 7 cm is rotated about the longer side. Find The total surface area of the solid thus generated.
S o l u t i o n :
Since the rectangular strip of 25 cm × 7 cm is rotated about the longer side , we have: Height, h = 25 cmRadius, r = 7cm ? Total surface area = 2 pr(r+h) = 2 p(7)(25+7) = 14 p(32) = 448 p
Q u e s t i o n : 6
A rectangular sheet of paper, 44 cm × 20 cm, is rolled along its length to form a cylinder. Find the total surface area of the cylinder thus generated.
S o l u t i o n :
The rectangular sheet of paper 44 cm ×20 cm is rolled along its length to form a cylinder.  The height of the cylinder is 20 cm and circumference is 44 cm.  We have: Height, h = 20 cmCircumference
Q u e s t i o n : 7
The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their curved surface areas.
S o l u t i o n :
Let the radii of two cylinders be 2r and 3r, respectively, and their heights be 5h and 3h, respectively. Let S
1
 and S
2
 be the curved surface areas of the two cylinder. S
1
= Curved surface
Q u e s t i o n : 8
The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. Prove that its height and radius are equal.
S o l u t i o n :
Let S
1
 and S
2
 be the curved surface area and total surface area of the circular cylinder, respectively. Then, S
1
= 2 prh , S
2
= 2 pr(r+h)According to the question:                   S
1
: S
2
= 1: 22 p
Q u e s t i o n : 9
The curved surface area of a cylinder is 1320 cm
2
 and its base has diameter 21 cm. Find the height of the cylinder.
S o l u t i o n :
Let h be the height of the cylinder. Given: Curved surface area, S = 1320 cm
2
 Diameter, d = 21 cmRadius, r = 10. 5     S = 2 prh1320 = 2 p ×10. 5 ×h     h =
1320
2 p ×10.5
     h = 20 cm
Q u e s t i o n : 1 0
The height of a right circular cylinder is 10.5 cm. If three times the sum of the areas of its two circular faces is twice the area of the curved surface area. Find the radius of its base.
S o l u t i o n :
Let r be the radius of the circular cylinder. Height, h = 10. 5 cmArea of the curved surface, S
1
= 2 prhSum of the areas of its two circular faces, S
2
= 2 pr
2
According to question:       3S
2
=
Q u e s t i o n : 1 1
Find the cost of plastering the inner surface of a well at Rs 9.50 per m
2
, if it is 21 m deep and diameter of its top is 6 m.
S o l u t i o n :
Given: Height, h = 21 mDiameter, d = 6 mRadius, r = 3 mArea of the inner surface of the well, S = 2 prh = 2 p ×3 ×21 m
2
= 2 ×
22
7
×3 ×21 m
2
= 396 m
2
According to question, the cost per m
Q u e s t i o n : 1 2
A cylindrical vessel open at the top has diameter 20 cm and height 14 cm. Find the cost of tin-plating it on the inside at the rate of 50 paise per hundred square centimetre.
S o l u t i o n :
Given: Diameter, d = 20 cmRadius, r = 10 cmHeight, h = 14 cmArea inside the cylindrical vessel that is to be tin -plated = SS = 2 prh + pr
2
= 2 p ×10 ×14 + p×10
2
= 280 p +100 p = 380 ×
22
7
Q u e s t i o n : 1 3
The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find the cost of plastering its inner curved surface at Rs 4 per square metre.
S o l u t i o n :
Given: Inner diameter of the circular well = 3. 5 m ? Inner radius of the circular well, r = 1. 75 mDepth of the circular well, h = 10 mInner curved surface area, S = 2 prhS = 2 p ×1. 75 ×10 m
Page 2


Q u e s t i o n : 1
Find the curved surface area and total surface area of a cylinder, the diameter of whose base is 7 cm and height is 60 cm.
S o l u t i o n :
Let r and h be the radius and the height of the cylinder. Given: r =
7
2
 cmh = 60 cmCurved surface area of the cylinder = 2 p ×r×h                                               = 2 ×
22
7
×
7
2
×60                          
Q u e s t i o n : 2
The curved surface area of a cylindrical road is 132 cm
2
. Find its length if the radius is 0.35 cm.
S o l u t i o n :
Consider h to be the height of the cylindrical rod. Given: Radius, r = 0. 35 cmCurved surface area = 132 cm
2
We know: Curved surface area = 2 × p ×r×h                       132 = 2 ×
22
7
×0. 35 ×h 
Q u e s t i o n : 3
The area of the base of a right circular cylinder is 616 cm
2
 and its height is 2.5 cm. Find the curved surface area of the cylinder.
S o l u t i o n :
Given: Area of the base of a right circular cylinder = 616 cm
2
Height = 2 .5 cm Let r be the radius of the base of a right circular cylinder. pr
2
 = 616 ? r
2
= 616 ×
7
22
? r
2
= 196 ? r = 14 cm
Q u e s t i o n : 4
The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find its curved surface area and total surface area.
S o l u t i o n :
Given: Height, h = 15 cmCircumference of the base of the cylinder = 88 cm
2
Let r be the radius of the cylinder. The circumference of the base of the cylinder = 2 pr88 = 2 ×
22
7
×rr =
88×7
2×22
= 14 
Q u e s t i o n : 5
A rectangular strip 25 cm × 7 cm is rotated about the longer side. Find The total surface area of the solid thus generated.
S o l u t i o n :
Since the rectangular strip of 25 cm × 7 cm is rotated about the longer side , we have: Height, h = 25 cmRadius, r = 7cm ? Total surface area = 2 pr(r+h) = 2 p(7)(25+7) = 14 p(32) = 448 p
Q u e s t i o n : 6
A rectangular sheet of paper, 44 cm × 20 cm, is rolled along its length to form a cylinder. Find the total surface area of the cylinder thus generated.
S o l u t i o n :
The rectangular sheet of paper 44 cm ×20 cm is rolled along its length to form a cylinder.  The height of the cylinder is 20 cm and circumference is 44 cm.  We have: Height, h = 20 cmCircumference
Q u e s t i o n : 7
The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their curved surface areas.
S o l u t i o n :
Let the radii of two cylinders be 2r and 3r, respectively, and their heights be 5h and 3h, respectively. Let S
1
 and S
2
 be the curved surface areas of the two cylinder. S
1
= Curved surface
Q u e s t i o n : 8
The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. Prove that its height and radius are equal.
S o l u t i o n :
Let S
1
 and S
2
 be the curved surface area and total surface area of the circular cylinder, respectively. Then, S
1
= 2 prh , S
2
= 2 pr(r+h)According to the question:                   S
1
: S
2
= 1: 22 p
Q u e s t i o n : 9
The curved surface area of a cylinder is 1320 cm
2
 and its base has diameter 21 cm. Find the height of the cylinder.
S o l u t i o n :
Let h be the height of the cylinder. Given: Curved surface area, S = 1320 cm
2
 Diameter, d = 21 cmRadius, r = 10. 5     S = 2 prh1320 = 2 p ×10. 5 ×h     h =
1320
2 p ×10.5
     h = 20 cm
Q u e s t i o n : 1 0
The height of a right circular cylinder is 10.5 cm. If three times the sum of the areas of its two circular faces is twice the area of the curved surface area. Find the radius of its base.
S o l u t i o n :
Let r be the radius of the circular cylinder. Height, h = 10. 5 cmArea of the curved surface, S
1
= 2 prhSum of the areas of its two circular faces, S
2
= 2 pr
2
According to question:       3S
2
=
Q u e s t i o n : 1 1
Find the cost of plastering the inner surface of a well at Rs 9.50 per m
2
, if it is 21 m deep and diameter of its top is 6 m.
S o l u t i o n :
Given: Height, h = 21 mDiameter, d = 6 mRadius, r = 3 mArea of the inner surface of the well, S = 2 prh = 2 p ×3 ×21 m
2
= 2 ×
22
7
×3 ×21 m
2
= 396 m
2
According to question, the cost per m
Q u e s t i o n : 1 2
A cylindrical vessel open at the top has diameter 20 cm and height 14 cm. Find the cost of tin-plating it on the inside at the rate of 50 paise per hundred square centimetre.
S o l u t i o n :
Given: Diameter, d = 20 cmRadius, r = 10 cmHeight, h = 14 cmArea inside the cylindrical vessel that is to be tin -plated = SS = 2 prh + pr
2
= 2 p ×10 ×14 + p×10
2
= 280 p +100 p = 380 ×
22
7
Q u e s t i o n : 1 3
The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find the cost of plastering its inner curved surface at Rs 4 per square metre.
S o l u t i o n :
Given: Inner diameter of the circular well = 3. 5 m ? Inner radius of the circular well, r = 1. 75 mDepth of the circular well, h = 10 mInner curved surface area, S = 2 prhS = 2 p ×1. 75 ×10 m
Q u e s t i o n : 1 4
The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions moving once over to level a playground. What is the area of the playground?
S o l u t i o n :
Given: Diameter of the roller = 84 cm ? Radius, r =
Diameter
2
= 42 cmIn 1 revolution, it covers the distance of its lateral surface area. Roller is a cylinder of height,  h = 120 cm  Radius = 42
Q u e s t i o n : 1 5
Twenty one cylindrical pillars of the Parliament House are to be cleaned. If the diameter of each pillar is 0.50 m and height is 4 m, what will be the cost of cleaning them at the rate of Rs
2.50 per square metre?
S o l u t i o n :
Given: Diameter of the pillars = 0. 5 mRadius of the pillars, r = 0. 25 mHeight of the pillars, h = 4 mNumber of pillars = 21Rate of cleaning = Rs 2. 50 per square metreCurved surface area
Q u e s t i o n : 1 6
The total surface area of a hollow cylinder which is open from both sides is 4620 sq. cm, area of base ring is 115.5 sq. cm and height 7 cm. Find the thickness of the cylinder.
S o l u t i o n :
Given: Total surface area of the cylinder = 4620 cm
2
Area of the base ring = 115. 5 cm
2
Height, h = 7 cmLet R be the radius of the outer ring and r be the radius of the inner ring. Area of the 
Q u e s t i o n : 1 7
The sum of the radius of the base and height of a solid cylinder is 37 m. If the total surface area of the solid cylinder is 1628 m
2
, find the circumference of its base.
S o l u t i o n :
Let r and h be the radius and height of the solid cylinder. Given: r+h = 37 mTotal surface area, S = 2 pr(r+h)1628 = 2 p ×r×37        r =
1628
2 p ×37
          =
1628
232.477
          = 7 mCircumference of its base
Q u e s t i o n : 1 8
Find the ratio between the total surface area of a cylinder to its curved surface area, given that its height and radius are 7.5 cm and 3.5 cm.
S o l u t i o n :
Let S
1
 and S
2
 be the total surface area and curved surface area, respectively. Given: Height, h = 7. 5 cmRadius, r = 3. 5 cmS
1
= 2 pr(r+h)S
2
= 2 prhAccording to the question:
S 1
S 2
=
2 p r(r+h)
2 p rh
S 1
S 2
=
Q u e s t i o n : 1 9
A cylindrical vessel, without lid, has to be tin-coated on its both sides. If the radius of the base is 70 cm and its height is 1.4 m, calculate the cost of tin-coating at the rate of Rs 3.50 per
1000 cm
2
.
S o l u t i o n :
Let r cm and h cm be the radius of the cylindrical vessel. Given: Radius, r = 70 cmHeight, h = 1. 4 m = 140 cmRate of tin -plating = Rs 3. 50 per 1000 square centimetreCost of tin -plating
Q u e s t i o n : 2 0
Find the volume of a cylinder whose
i
r = 3.5 cm, h = 40 cm
ii
r = 2.8 m, h = 15 m
S o l u t i o n :
i Given : r = 3. 5 cm, h = 40 cmVolume of cylinder, V = pr
2
h  =
22
7
×(3. 5)
2
×40  = 1540 cm
3
ii Given: r = 2. 8 m, h = 15 mVolume of cylinder, V = pr
2
h  =
22
7
×(2. 8)
2
×15  = 369. 6 m
3
Q u e s t i o n : 2 1
Find the volume of a cylinder, if the diameter (d) of its base and its altitude (h) are:
i
d = 21 cm, h = 10 cm
ii
d = 7 m, h = 24 m
S o l u t i o n :
i Given: d = 21 cm, radius, r =
d
2
= 10. 5 cmheight, h = 10 cmVolume of the cylinder, V = pr
2
h =
22
7
×(10. 5)
2
×10 = 3465 cm
3
ii Given: d = 7 m, radius, r =
d
2
= 3. 5 mheight h = 24 mVolume
Q u e s t i o n : 2 2
The area of the base of a right circular cylinder is 616 cm
2
 and its height is 25 cm. Find the volume of the cylinder.
S o l u t i o n :
Let the area of the base of a right circular cylinder be S cm
2
. Given: S = 616 cm
2
Height, h = 25 cmLet the radius of a right circular cylinder be r cm.    S = pr
2
616 =
22
7
×r
2
 r
   2
=
616×7
22
  r
2
 = 196
Q u e s t i o n : 2 3
The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find the volume of the cylinder.
S o l u t i o n :
Let r cm be the radius of a cylinder. Circumference of the cylinder, S = 2 prGiven: Height, h = 15 cmCircumference, S = 88 cm S = 2 pr88 = 2 ×
22
7
×r  r =
88×7
44
  r = 14 cmVolume of cylinder, V
Q u e s t i o n : 2 4
A hollow cylindrical pipe is 21 dm long. Its outer and inner diameters are 10 cm and 6 cm respectively. Find the volume of the copper used in making the pipe.
S o l u t i o n :
Let the length of the cylinder pipe be h = 21, dm = 210 cm. Let the outer and the inner radius of the pipe be R cm and r cm, respectively. ? 2R = 10  and 2r = 6R = 5 cm and r = 3cmVolume
Q u e s t i o n : 2 5
Find the i
( ) ( )
( ) ( )
Page 3


Q u e s t i o n : 1
Find the curved surface area and total surface area of a cylinder, the diameter of whose base is 7 cm and height is 60 cm.
S o l u t i o n :
Let r and h be the radius and the height of the cylinder. Given: r =
7
2
 cmh = 60 cmCurved surface area of the cylinder = 2 p ×r×h                                               = 2 ×
22
7
×
7
2
×60                          
Q u e s t i o n : 2
The curved surface area of a cylindrical road is 132 cm
2
. Find its length if the radius is 0.35 cm.
S o l u t i o n :
Consider h to be the height of the cylindrical rod. Given: Radius, r = 0. 35 cmCurved surface area = 132 cm
2
We know: Curved surface area = 2 × p ×r×h                       132 = 2 ×
22
7
×0. 35 ×h 
Q u e s t i o n : 3
The area of the base of a right circular cylinder is 616 cm
2
 and its height is 2.5 cm. Find the curved surface area of the cylinder.
S o l u t i o n :
Given: Area of the base of a right circular cylinder = 616 cm
2
Height = 2 .5 cm Let r be the radius of the base of a right circular cylinder. pr
2
 = 616 ? r
2
= 616 ×
7
22
? r
2
= 196 ? r = 14 cm
Q u e s t i o n : 4
The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find its curved surface area and total surface area.
S o l u t i o n :
Given: Height, h = 15 cmCircumference of the base of the cylinder = 88 cm
2
Let r be the radius of the cylinder. The circumference of the base of the cylinder = 2 pr88 = 2 ×
22
7
×rr =
88×7
2×22
= 14 
Q u e s t i o n : 5
A rectangular strip 25 cm × 7 cm is rotated about the longer side. Find The total surface area of the solid thus generated.
S o l u t i o n :
Since the rectangular strip of 25 cm × 7 cm is rotated about the longer side , we have: Height, h = 25 cmRadius, r = 7cm ? Total surface area = 2 pr(r+h) = 2 p(7)(25+7) = 14 p(32) = 448 p
Q u e s t i o n : 6
A rectangular sheet of paper, 44 cm × 20 cm, is rolled along its length to form a cylinder. Find the total surface area of the cylinder thus generated.
S o l u t i o n :
The rectangular sheet of paper 44 cm ×20 cm is rolled along its length to form a cylinder.  The height of the cylinder is 20 cm and circumference is 44 cm.  We have: Height, h = 20 cmCircumference
Q u e s t i o n : 7
The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their curved surface areas.
S o l u t i o n :
Let the radii of two cylinders be 2r and 3r, respectively, and their heights be 5h and 3h, respectively. Let S
1
 and S
2
 be the curved surface areas of the two cylinder. S
1
= Curved surface
Q u e s t i o n : 8
The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. Prove that its height and radius are equal.
S o l u t i o n :
Let S
1
 and S
2
 be the curved surface area and total surface area of the circular cylinder, respectively. Then, S
1
= 2 prh , S
2
= 2 pr(r+h)According to the question:                   S
1
: S
2
= 1: 22 p
Q u e s t i o n : 9
The curved surface area of a cylinder is 1320 cm
2
 and its base has diameter 21 cm. Find the height of the cylinder.
S o l u t i o n :
Let h be the height of the cylinder. Given: Curved surface area, S = 1320 cm
2
 Diameter, d = 21 cmRadius, r = 10. 5     S = 2 prh1320 = 2 p ×10. 5 ×h     h =
1320
2 p ×10.5
     h = 20 cm
Q u e s t i o n : 1 0
The height of a right circular cylinder is 10.5 cm. If three times the sum of the areas of its two circular faces is twice the area of the curved surface area. Find the radius of its base.
S o l u t i o n :
Let r be the radius of the circular cylinder. Height, h = 10. 5 cmArea of the curved surface, S
1
= 2 prhSum of the areas of its two circular faces, S
2
= 2 pr
2
According to question:       3S
2
=
Q u e s t i o n : 1 1
Find the cost of plastering the inner surface of a well at Rs 9.50 per m
2
, if it is 21 m deep and diameter of its top is 6 m.
S o l u t i o n :
Given: Height, h = 21 mDiameter, d = 6 mRadius, r = 3 mArea of the inner surface of the well, S = 2 prh = 2 p ×3 ×21 m
2
= 2 ×
22
7
×3 ×21 m
2
= 396 m
2
According to question, the cost per m
Q u e s t i o n : 1 2
A cylindrical vessel open at the top has diameter 20 cm and height 14 cm. Find the cost of tin-plating it on the inside at the rate of 50 paise per hundred square centimetre.
S o l u t i o n :
Given: Diameter, d = 20 cmRadius, r = 10 cmHeight, h = 14 cmArea inside the cylindrical vessel that is to be tin -plated = SS = 2 prh + pr
2
= 2 p ×10 ×14 + p×10
2
= 280 p +100 p = 380 ×
22
7
Q u e s t i o n : 1 3
The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find the cost of plastering its inner curved surface at Rs 4 per square metre.
S o l u t i o n :
Given: Inner diameter of the circular well = 3. 5 m ? Inner radius of the circular well, r = 1. 75 mDepth of the circular well, h = 10 mInner curved surface area, S = 2 prhS = 2 p ×1. 75 ×10 m
Q u e s t i o n : 1 4
The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions moving once over to level a playground. What is the area of the playground?
S o l u t i o n :
Given: Diameter of the roller = 84 cm ? Radius, r =
Diameter
2
= 42 cmIn 1 revolution, it covers the distance of its lateral surface area. Roller is a cylinder of height,  h = 120 cm  Radius = 42
Q u e s t i o n : 1 5
Twenty one cylindrical pillars of the Parliament House are to be cleaned. If the diameter of each pillar is 0.50 m and height is 4 m, what will be the cost of cleaning them at the rate of Rs
2.50 per square metre?
S o l u t i o n :
Given: Diameter of the pillars = 0. 5 mRadius of the pillars, r = 0. 25 mHeight of the pillars, h = 4 mNumber of pillars = 21Rate of cleaning = Rs 2. 50 per square metreCurved surface area
Q u e s t i o n : 1 6
The total surface area of a hollow cylinder which is open from both sides is 4620 sq. cm, area of base ring is 115.5 sq. cm and height 7 cm. Find the thickness of the cylinder.
S o l u t i o n :
Given: Total surface area of the cylinder = 4620 cm
2
Area of the base ring = 115. 5 cm
2
Height, h = 7 cmLet R be the radius of the outer ring and r be the radius of the inner ring. Area of the 
Q u e s t i o n : 1 7
The sum of the radius of the base and height of a solid cylinder is 37 m. If the total surface area of the solid cylinder is 1628 m
2
, find the circumference of its base.
S o l u t i o n :
Let r and h be the radius and height of the solid cylinder. Given: r+h = 37 mTotal surface area, S = 2 pr(r+h)1628 = 2 p ×r×37        r =
1628
2 p ×37
          =
1628
232.477
          = 7 mCircumference of its base
Q u e s t i o n : 1 8
Find the ratio between the total surface area of a cylinder to its curved surface area, given that its height and radius are 7.5 cm and 3.5 cm.
S o l u t i o n :
Let S
1
 and S
2
 be the total surface area and curved surface area, respectively. Given: Height, h = 7. 5 cmRadius, r = 3. 5 cmS
1
= 2 pr(r+h)S
2
= 2 prhAccording to the question:
S 1
S 2
=
2 p r(r+h)
2 p rh
S 1
S 2
=
Q u e s t i o n : 1 9
A cylindrical vessel, without lid, has to be tin-coated on its both sides. If the radius of the base is 70 cm and its height is 1.4 m, calculate the cost of tin-coating at the rate of Rs 3.50 per
1000 cm
2
.
S o l u t i o n :
Let r cm and h cm be the radius of the cylindrical vessel. Given: Radius, r = 70 cmHeight, h = 1. 4 m = 140 cmRate of tin -plating = Rs 3. 50 per 1000 square centimetreCost of tin -plating
Q u e s t i o n : 2 0
Find the volume of a cylinder whose
i
r = 3.5 cm, h = 40 cm
ii
r = 2.8 m, h = 15 m
S o l u t i o n :
i Given : r = 3. 5 cm, h = 40 cmVolume of cylinder, V = pr
2
h  =
22
7
×(3. 5)
2
×40  = 1540 cm
3
ii Given: r = 2. 8 m, h = 15 mVolume of cylinder, V = pr
2
h  =
22
7
×(2. 8)
2
×15  = 369. 6 m
3
Q u e s t i o n : 2 1
Find the volume of a cylinder, if the diameter (d) of its base and its altitude (h) are:
i
d = 21 cm, h = 10 cm
ii
d = 7 m, h = 24 m
S o l u t i o n :
i Given: d = 21 cm, radius, r =
d
2
= 10. 5 cmheight, h = 10 cmVolume of the cylinder, V = pr
2
h =
22
7
×(10. 5)
2
×10 = 3465 cm
3
ii Given: d = 7 m, radius, r =
d
2
= 3. 5 mheight h = 24 mVolume
Q u e s t i o n : 2 2
The area of the base of a right circular cylinder is 616 cm
2
 and its height is 25 cm. Find the volume of the cylinder.
S o l u t i o n :
Let the area of the base of a right circular cylinder be S cm
2
. Given: S = 616 cm
2
Height, h = 25 cmLet the radius of a right circular cylinder be r cm.    S = pr
2
616 =
22
7
×r
2
 r
   2
=
616×7
22
  r
2
 = 196
Q u e s t i o n : 2 3
The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find the volume of the cylinder.
S o l u t i o n :
Let r cm be the radius of a cylinder. Circumference of the cylinder, S = 2 prGiven: Height, h = 15 cmCircumference, S = 88 cm S = 2 pr88 = 2 ×
22
7
×r  r =
88×7
44
  r = 14 cmVolume of cylinder, V
Q u e s t i o n : 2 4
A hollow cylindrical pipe is 21 dm long. Its outer and inner diameters are 10 cm and 6 cm respectively. Find the volume of the copper used in making the pipe.
S o l u t i o n :
Let the length of the cylinder pipe be h = 21, dm = 210 cm. Let the outer and the inner radius of the pipe be R cm and r cm, respectively. ? 2R = 10  and 2r = 6R = 5 cm and r = 3cmVolume
Q u e s t i o n : 2 5
Find the i
( ) ( )
( ) ( )
curved surface area ii
total surface area and iii
volume of a right circular cylinder whose height is 15 cm and the radius of the base is 7 cm.
S o l u t i o n :
Given: Height, h = 15 cmRadius, r = 7 cm i Curved surface area, S
1
= 2 prh   = 2 ×
22
7
×7 ×15   = 660 cm
2
ii Total surface area, S
2
= 2 pr(r+h)   = 2 ×
22
7
×7 ×(7 +15)   = 44 ×22   = 968
Q u e s t i o n : 2 6
The diameter of the base of a right circular cylinder is 42 cm and its height is 10 cm. Find the volume of the cylinder.
S o l u t i o n :
Given: Diameter, d = 42 cmRadius, r =
d
2
= 21 cmHeight, h = 10 cmVolume of the cylinder, V = pr
2
h  =
22
7
×21
2
×10  = 13860 cm
3
Q u e s t i o n : 2 7
Find the volume of a cylinder, the diameter of whose base is 7 cm and height being 60 cm. Also, find the capacity of the cylinder in litres.
S o l u t i o n :
Given: Diameter, d = 7 cmRadius, r = 3. 5 cmHeight, h = 60 cmVolume of the cylinder, V = pr
2
h  =
22
7
×3. 5
2
×60  = 2310 cm
3
Capacity of the cylinder in litres =
2310
1000
           ( 1litre = 1000
Q u e s t i o n : 2 8
A rectangular strip 25 cm × 7 cm is rotated about the longer side. Find the volume of the solid, thus generated.
S o l u t i o n :
Given: Rectangular strip has radius, r = 7 cmHeight, h = 25 cmVolume of the solid, V = pr
2
h  =
22
7
×7
2
×25  = 3850 cm
3
Q u e s t i o n : 2 9
A rectangular sheet of paper, 44 cm × 20 cm, is rolled along its length to form a cylinder. Find the volume of the cylinder so formed.
S o l u t i o n :
The length l and breadth b of the rectangular sheet are 44 cm and 20 cmNow, the sheet is rolled along the length to form a cylinder. Let the radius of the cylinder be r cm. Height, h =
Q u e s t i o n : 3 0
The volume and the curved surface area of a cylinder are 1650 cm
3
 and 660 cm
2
 respectively. Find the radius and height of the cylinder.
S o l u t i o n :
Curved surface area of the cylinder = 2 prh  =660 cm
2
     ... 1
Volume of the cylinder = pr
2
h   =1650 cm
3
                       ... 2
From 1
and 2
, we can calculate the radius (r) and the height of cylinder (h).
We know the volume of the cylinder, i.e. 1650 cm
3
?  1650 = pr
2
h
     
   h = 
1650
p r
2
Substituting h into 1
:
660 = 2 prh
660 = 2 pr × 
1650
p r
2
660r = 21650
r = 5 cm
h = 
1650
p r
2
     = 
1650
22
7
×5
2
 = 21 cm.
Hence, the radius and the height of the cylinder are 5 cm and 21 cm, respectively.
Q u e s t i o n : 3 1
The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their volumes.
S o l u t i o n :
Here, r
1
 = Radius of cylinder 1
         h
1
 = Height of cylinder 1
         r
2
 = Radius of cylinder 2
         h
2
 = Height of cylinder 2
         V
1
 = Volume of cylinder 1
         V
2
 = Volume of cylinder 2
Ratio of the radii of two cylinders = 2:3
Ratio of the heights of two cylinders = 5:3
Volume of the cylinder =  pr
2
h
V
1
/V
2
 = ( pr
1
2
h
1
)/( pr
2
2
h
2
) = ( p(2r)
2
5h)/( p(3r)
2
3h) 
V
1
/V
2
 = ( p4r
2
5h)/( p9r
2
3h) = 20 / 27
Hence, the ratio of their volumes is 20:27
Q u e s t i o n : 3 2
( ) ( )
( ) ( )
Page 4


Q u e s t i o n : 1
Find the curved surface area and total surface area of a cylinder, the diameter of whose base is 7 cm and height is 60 cm.
S o l u t i o n :
Let r and h be the radius and the height of the cylinder. Given: r =
7
2
 cmh = 60 cmCurved surface area of the cylinder = 2 p ×r×h                                               = 2 ×
22
7
×
7
2
×60                          
Q u e s t i o n : 2
The curved surface area of a cylindrical road is 132 cm
2
. Find its length if the radius is 0.35 cm.
S o l u t i o n :
Consider h to be the height of the cylindrical rod. Given: Radius, r = 0. 35 cmCurved surface area = 132 cm
2
We know: Curved surface area = 2 × p ×r×h                       132 = 2 ×
22
7
×0. 35 ×h 
Q u e s t i o n : 3
The area of the base of a right circular cylinder is 616 cm
2
 and its height is 2.5 cm. Find the curved surface area of the cylinder.
S o l u t i o n :
Given: Area of the base of a right circular cylinder = 616 cm
2
Height = 2 .5 cm Let r be the radius of the base of a right circular cylinder. pr
2
 = 616 ? r
2
= 616 ×
7
22
? r
2
= 196 ? r = 14 cm
Q u e s t i o n : 4
The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find its curved surface area and total surface area.
S o l u t i o n :
Given: Height, h = 15 cmCircumference of the base of the cylinder = 88 cm
2
Let r be the radius of the cylinder. The circumference of the base of the cylinder = 2 pr88 = 2 ×
22
7
×rr =
88×7
2×22
= 14 
Q u e s t i o n : 5
A rectangular strip 25 cm × 7 cm is rotated about the longer side. Find The total surface area of the solid thus generated.
S o l u t i o n :
Since the rectangular strip of 25 cm × 7 cm is rotated about the longer side , we have: Height, h = 25 cmRadius, r = 7cm ? Total surface area = 2 pr(r+h) = 2 p(7)(25+7) = 14 p(32) = 448 p
Q u e s t i o n : 6
A rectangular sheet of paper, 44 cm × 20 cm, is rolled along its length to form a cylinder. Find the total surface area of the cylinder thus generated.
S o l u t i o n :
The rectangular sheet of paper 44 cm ×20 cm is rolled along its length to form a cylinder.  The height of the cylinder is 20 cm and circumference is 44 cm.  We have: Height, h = 20 cmCircumference
Q u e s t i o n : 7
The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their curved surface areas.
S o l u t i o n :
Let the radii of two cylinders be 2r and 3r, respectively, and their heights be 5h and 3h, respectively. Let S
1
 and S
2
 be the curved surface areas of the two cylinder. S
1
= Curved surface
Q u e s t i o n : 8
The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. Prove that its height and radius are equal.
S o l u t i o n :
Let S
1
 and S
2
 be the curved surface area and total surface area of the circular cylinder, respectively. Then, S
1
= 2 prh , S
2
= 2 pr(r+h)According to the question:                   S
1
: S
2
= 1: 22 p
Q u e s t i o n : 9
The curved surface area of a cylinder is 1320 cm
2
 and its base has diameter 21 cm. Find the height of the cylinder.
S o l u t i o n :
Let h be the height of the cylinder. Given: Curved surface area, S = 1320 cm
2
 Diameter, d = 21 cmRadius, r = 10. 5     S = 2 prh1320 = 2 p ×10. 5 ×h     h =
1320
2 p ×10.5
     h = 20 cm
Q u e s t i o n : 1 0
The height of a right circular cylinder is 10.5 cm. If three times the sum of the areas of its two circular faces is twice the area of the curved surface area. Find the radius of its base.
S o l u t i o n :
Let r be the radius of the circular cylinder. Height, h = 10. 5 cmArea of the curved surface, S
1
= 2 prhSum of the areas of its two circular faces, S
2
= 2 pr
2
According to question:       3S
2
=
Q u e s t i o n : 1 1
Find the cost of plastering the inner surface of a well at Rs 9.50 per m
2
, if it is 21 m deep and diameter of its top is 6 m.
S o l u t i o n :
Given: Height, h = 21 mDiameter, d = 6 mRadius, r = 3 mArea of the inner surface of the well, S = 2 prh = 2 p ×3 ×21 m
2
= 2 ×
22
7
×3 ×21 m
2
= 396 m
2
According to question, the cost per m
Q u e s t i o n : 1 2
A cylindrical vessel open at the top has diameter 20 cm and height 14 cm. Find the cost of tin-plating it on the inside at the rate of 50 paise per hundred square centimetre.
S o l u t i o n :
Given: Diameter, d = 20 cmRadius, r = 10 cmHeight, h = 14 cmArea inside the cylindrical vessel that is to be tin -plated = SS = 2 prh + pr
2
= 2 p ×10 ×14 + p×10
2
= 280 p +100 p = 380 ×
22
7
Q u e s t i o n : 1 3
The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find the cost of plastering its inner curved surface at Rs 4 per square metre.
S o l u t i o n :
Given: Inner diameter of the circular well = 3. 5 m ? Inner radius of the circular well, r = 1. 75 mDepth of the circular well, h = 10 mInner curved surface area, S = 2 prhS = 2 p ×1. 75 ×10 m
Q u e s t i o n : 1 4
The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions moving once over to level a playground. What is the area of the playground?
S o l u t i o n :
Given: Diameter of the roller = 84 cm ? Radius, r =
Diameter
2
= 42 cmIn 1 revolution, it covers the distance of its lateral surface area. Roller is a cylinder of height,  h = 120 cm  Radius = 42
Q u e s t i o n : 1 5
Twenty one cylindrical pillars of the Parliament House are to be cleaned. If the diameter of each pillar is 0.50 m and height is 4 m, what will be the cost of cleaning them at the rate of Rs
2.50 per square metre?
S o l u t i o n :
Given: Diameter of the pillars = 0. 5 mRadius of the pillars, r = 0. 25 mHeight of the pillars, h = 4 mNumber of pillars = 21Rate of cleaning = Rs 2. 50 per square metreCurved surface area
Q u e s t i o n : 1 6
The total surface area of a hollow cylinder which is open from both sides is 4620 sq. cm, area of base ring is 115.5 sq. cm and height 7 cm. Find the thickness of the cylinder.
S o l u t i o n :
Given: Total surface area of the cylinder = 4620 cm
2
Area of the base ring = 115. 5 cm
2
Height, h = 7 cmLet R be the radius of the outer ring and r be the radius of the inner ring. Area of the 
Q u e s t i o n : 1 7
The sum of the radius of the base and height of a solid cylinder is 37 m. If the total surface area of the solid cylinder is 1628 m
2
, find the circumference of its base.
S o l u t i o n :
Let r and h be the radius and height of the solid cylinder. Given: r+h = 37 mTotal surface area, S = 2 pr(r+h)1628 = 2 p ×r×37        r =
1628
2 p ×37
          =
1628
232.477
          = 7 mCircumference of its base
Q u e s t i o n : 1 8
Find the ratio between the total surface area of a cylinder to its curved surface area, given that its height and radius are 7.5 cm and 3.5 cm.
S o l u t i o n :
Let S
1
 and S
2
 be the total surface area and curved surface area, respectively. Given: Height, h = 7. 5 cmRadius, r = 3. 5 cmS
1
= 2 pr(r+h)S
2
= 2 prhAccording to the question:
S 1
S 2
=
2 p r(r+h)
2 p rh
S 1
S 2
=
Q u e s t i o n : 1 9
A cylindrical vessel, without lid, has to be tin-coated on its both sides. If the radius of the base is 70 cm and its height is 1.4 m, calculate the cost of tin-coating at the rate of Rs 3.50 per
1000 cm
2
.
S o l u t i o n :
Let r cm and h cm be the radius of the cylindrical vessel. Given: Radius, r = 70 cmHeight, h = 1. 4 m = 140 cmRate of tin -plating = Rs 3. 50 per 1000 square centimetreCost of tin -plating
Q u e s t i o n : 2 0
Find the volume of a cylinder whose
i
r = 3.5 cm, h = 40 cm
ii
r = 2.8 m, h = 15 m
S o l u t i o n :
i Given : r = 3. 5 cm, h = 40 cmVolume of cylinder, V = pr
2
h  =
22
7
×(3. 5)
2
×40  = 1540 cm
3
ii Given: r = 2. 8 m, h = 15 mVolume of cylinder, V = pr
2
h  =
22
7
×(2. 8)
2
×15  = 369. 6 m
3
Q u e s t i o n : 2 1
Find the volume of a cylinder, if the diameter (d) of its base and its altitude (h) are:
i
d = 21 cm, h = 10 cm
ii
d = 7 m, h = 24 m
S o l u t i o n :
i Given: d = 21 cm, radius, r =
d
2
= 10. 5 cmheight, h = 10 cmVolume of the cylinder, V = pr
2
h =
22
7
×(10. 5)
2
×10 = 3465 cm
3
ii Given: d = 7 m, radius, r =
d
2
= 3. 5 mheight h = 24 mVolume
Q u e s t i o n : 2 2
The area of the base of a right circular cylinder is 616 cm
2
 and its height is 25 cm. Find the volume of the cylinder.
S o l u t i o n :
Let the area of the base of a right circular cylinder be S cm
2
. Given: S = 616 cm
2
Height, h = 25 cmLet the radius of a right circular cylinder be r cm.    S = pr
2
616 =
22
7
×r
2
 r
   2
=
616×7
22
  r
2
 = 196
Q u e s t i o n : 2 3
The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find the volume of the cylinder.
S o l u t i o n :
Let r cm be the radius of a cylinder. Circumference of the cylinder, S = 2 prGiven: Height, h = 15 cmCircumference, S = 88 cm S = 2 pr88 = 2 ×
22
7
×r  r =
88×7
44
  r = 14 cmVolume of cylinder, V
Q u e s t i o n : 2 4
A hollow cylindrical pipe is 21 dm long. Its outer and inner diameters are 10 cm and 6 cm respectively. Find the volume of the copper used in making the pipe.
S o l u t i o n :
Let the length of the cylinder pipe be h = 21, dm = 210 cm. Let the outer and the inner radius of the pipe be R cm and r cm, respectively. ? 2R = 10  and 2r = 6R = 5 cm and r = 3cmVolume
Q u e s t i o n : 2 5
Find the i
( ) ( )
( ) ( )
curved surface area ii
total surface area and iii
volume of a right circular cylinder whose height is 15 cm and the radius of the base is 7 cm.
S o l u t i o n :
Given: Height, h = 15 cmRadius, r = 7 cm i Curved surface area, S
1
= 2 prh   = 2 ×
22
7
×7 ×15   = 660 cm
2
ii Total surface area, S
2
= 2 pr(r+h)   = 2 ×
22
7
×7 ×(7 +15)   = 44 ×22   = 968
Q u e s t i o n : 2 6
The diameter of the base of a right circular cylinder is 42 cm and its height is 10 cm. Find the volume of the cylinder.
S o l u t i o n :
Given: Diameter, d = 42 cmRadius, r =
d
2
= 21 cmHeight, h = 10 cmVolume of the cylinder, V = pr
2
h  =
22
7
×21
2
×10  = 13860 cm
3
Q u e s t i o n : 2 7
Find the volume of a cylinder, the diameter of whose base is 7 cm and height being 60 cm. Also, find the capacity of the cylinder in litres.
S o l u t i o n :
Given: Diameter, d = 7 cmRadius, r = 3. 5 cmHeight, h = 60 cmVolume of the cylinder, V = pr
2
h  =
22
7
×3. 5
2
×60  = 2310 cm
3
Capacity of the cylinder in litres =
2310
1000
           ( 1litre = 1000
Q u e s t i o n : 2 8
A rectangular strip 25 cm × 7 cm is rotated about the longer side. Find the volume of the solid, thus generated.
S o l u t i o n :
Given: Rectangular strip has radius, r = 7 cmHeight, h = 25 cmVolume of the solid, V = pr
2
h  =
22
7
×7
2
×25  = 3850 cm
3
Q u e s t i o n : 2 9
A rectangular sheet of paper, 44 cm × 20 cm, is rolled along its length to form a cylinder. Find the volume of the cylinder so formed.
S o l u t i o n :
The length l and breadth b of the rectangular sheet are 44 cm and 20 cmNow, the sheet is rolled along the length to form a cylinder. Let the radius of the cylinder be r cm. Height, h =
Q u e s t i o n : 3 0
The volume and the curved surface area of a cylinder are 1650 cm
3
 and 660 cm
2
 respectively. Find the radius and height of the cylinder.
S o l u t i o n :
Curved surface area of the cylinder = 2 prh  =660 cm
2
     ... 1
Volume of the cylinder = pr
2
h   =1650 cm
3
                       ... 2
From 1
and 2
, we can calculate the radius (r) and the height of cylinder (h).
We know the volume of the cylinder, i.e. 1650 cm
3
?  1650 = pr
2
h
     
   h = 
1650
p r
2
Substituting h into 1
:
660 = 2 prh
660 = 2 pr × 
1650
p r
2
660r = 21650
r = 5 cm
h = 
1650
p r
2
     = 
1650
22
7
×5
2
 = 21 cm.
Hence, the radius and the height of the cylinder are 5 cm and 21 cm, respectively.
Q u e s t i o n : 3 1
The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their volumes.
S o l u t i o n :
Here, r
1
 = Radius of cylinder 1
         h
1
 = Height of cylinder 1
         r
2
 = Radius of cylinder 2
         h
2
 = Height of cylinder 2
         V
1
 = Volume of cylinder 1
         V
2
 = Volume of cylinder 2
Ratio of the radii of two cylinders = 2:3
Ratio of the heights of two cylinders = 5:3
Volume of the cylinder =  pr
2
h
V
1
/V
2
 = ( pr
1
2
h
1
)/( pr
2
2
h
2
) = ( p(2r)
2
5h)/( p(3r)
2
3h) 
V
1
/V
2
 = ( p4r
2
5h)/( p9r
2
3h) = 20 / 27
Hence, the ratio of their volumes is 20:27
Q u e s t i o n : 3 2
( ) ( )
( ) ( )
The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. Find the volume of the cylinder, if its total surface area is 616 cm
2
.
S o l u t i o n :
Let r cm be the radius and h cm be the length of the cylinder. The curved surface area and the total surface area is 1:2.
The total surface area is 616 cm
2
.
The curved surface area is the half of 616 cm
2
, i.e. 308 cm
2
?.
Curved area = 2 prh
       So, h = 
308
2 p r
Total surface area = Curved surface area + Top and bottom area
Top and bottom area = 616 - 308 = 308 cm
2
 = 2 pr
2
r
2
 = 
308
2×
22
7
r = 7 cm
h = 
308
2× p ×7
= 7cm
Then, the volume of the cylinder can be calculated as follows:
V = 
22
7
× 7
2
 × 7 = 1078
Hence, it is obtained that the volume of the cylinder is 1078 cm
3
.
Q u e s t i o n : 3 3
The curved surface area of a cylinder is 1320 cm
2
 and its base has diameter 21 cm. Find the volume of the cylinder.
S o l u t i o n :
r cm = Radius of the cylinder
h cm = Height of the cylinder
Diameter of the cylinder is 21 cm. Thus, the radius is 10.5 cm.
Since the curved surface area has been known, we can calculate h by the equation given below:
The curved surface area of the cylinder = 2 prh
1320 cm
2
= 2 prh
1320 cm
2
= 2 x 22 x 10.5cm
x h
?                       7
h = 20 cm
? Volume of the cylinder (V) = pr
2
h
V = 2210.5cm
2
20cm
        7
V = 6930 cm
3
Q u e s t i o n : 3 4
The ratio between the radius of the base and the height of a cylinder is 2 : 3. Find the total surface area of the cylinder, if its volume is 1617 cm
3
.
S o l u t i o n :
Let r cm be the radius and h cm be the height of the cylinder. It is given that the ratio of r and h is 2:3, so h = 1.5r
The volume of the cylinder (V) is 1617 cm
3
.
So, we can find the radius and the height of the cylinder from the equation given below:
V= pr
2
h
1617 = pr
2
h
1617 = pr
2
(1.5r)
 r
3
 =343
r = 7 cm and h = 10.5 cm
Total surface area = 2 pr
2
+2 prh
?                                =2 ×
22
7
×7
2
+2 ×
22
7
×7 ×10. 5 = 770 cm
2
                              
Hence, the total surface area of the cylinder is 770 cm
2
.
Q u e s t i o n : 3 5
The curved surface area of a cylindrical pillar is 264 m
2
 and its volume is 924 m
3
. Find the diameter and the height of the pillar.
S o l u t i o n :
Here, r  m= radius of the cylinder
         h m= height of the cylinder
Curved surface area of the cylinder = 2 prh     ... 1
Volume of the cylinder = pr
2
h                        ... 2
            924 = pr
2
h
            h =
924
p r
2
Page 5


Q u e s t i o n : 1
Find the curved surface area and total surface area of a cylinder, the diameter of whose base is 7 cm and height is 60 cm.
S o l u t i o n :
Let r and h be the radius and the height of the cylinder. Given: r =
7
2
 cmh = 60 cmCurved surface area of the cylinder = 2 p ×r×h                                               = 2 ×
22
7
×
7
2
×60                          
Q u e s t i o n : 2
The curved surface area of a cylindrical road is 132 cm
2
. Find its length if the radius is 0.35 cm.
S o l u t i o n :
Consider h to be the height of the cylindrical rod. Given: Radius, r = 0. 35 cmCurved surface area = 132 cm
2
We know: Curved surface area = 2 × p ×r×h                       132 = 2 ×
22
7
×0. 35 ×h 
Q u e s t i o n : 3
The area of the base of a right circular cylinder is 616 cm
2
 and its height is 2.5 cm. Find the curved surface area of the cylinder.
S o l u t i o n :
Given: Area of the base of a right circular cylinder = 616 cm
2
Height = 2 .5 cm Let r be the radius of the base of a right circular cylinder. pr
2
 = 616 ? r
2
= 616 ×
7
22
? r
2
= 196 ? r = 14 cm
Q u e s t i o n : 4
The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find its curved surface area and total surface area.
S o l u t i o n :
Given: Height, h = 15 cmCircumference of the base of the cylinder = 88 cm
2
Let r be the radius of the cylinder. The circumference of the base of the cylinder = 2 pr88 = 2 ×
22
7
×rr =
88×7
2×22
= 14 
Q u e s t i o n : 5
A rectangular strip 25 cm × 7 cm is rotated about the longer side. Find The total surface area of the solid thus generated.
S o l u t i o n :
Since the rectangular strip of 25 cm × 7 cm is rotated about the longer side , we have: Height, h = 25 cmRadius, r = 7cm ? Total surface area = 2 pr(r+h) = 2 p(7)(25+7) = 14 p(32) = 448 p
Q u e s t i o n : 6
A rectangular sheet of paper, 44 cm × 20 cm, is rolled along its length to form a cylinder. Find the total surface area of the cylinder thus generated.
S o l u t i o n :
The rectangular sheet of paper 44 cm ×20 cm is rolled along its length to form a cylinder.  The height of the cylinder is 20 cm and circumference is 44 cm.  We have: Height, h = 20 cmCircumference
Q u e s t i o n : 7
The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their curved surface areas.
S o l u t i o n :
Let the radii of two cylinders be 2r and 3r, respectively, and their heights be 5h and 3h, respectively. Let S
1
 and S
2
 be the curved surface areas of the two cylinder. S
1
= Curved surface
Q u e s t i o n : 8
The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. Prove that its height and radius are equal.
S o l u t i o n :
Let S
1
 and S
2
 be the curved surface area and total surface area of the circular cylinder, respectively. Then, S
1
= 2 prh , S
2
= 2 pr(r+h)According to the question:                   S
1
: S
2
= 1: 22 p
Q u e s t i o n : 9
The curved surface area of a cylinder is 1320 cm
2
 and its base has diameter 21 cm. Find the height of the cylinder.
S o l u t i o n :
Let h be the height of the cylinder. Given: Curved surface area, S = 1320 cm
2
 Diameter, d = 21 cmRadius, r = 10. 5     S = 2 prh1320 = 2 p ×10. 5 ×h     h =
1320
2 p ×10.5
     h = 20 cm
Q u e s t i o n : 1 0
The height of a right circular cylinder is 10.5 cm. If three times the sum of the areas of its two circular faces is twice the area of the curved surface area. Find the radius of its base.
S o l u t i o n :
Let r be the radius of the circular cylinder. Height, h = 10. 5 cmArea of the curved surface, S
1
= 2 prhSum of the areas of its two circular faces, S
2
= 2 pr
2
According to question:       3S
2
=
Q u e s t i o n : 1 1
Find the cost of plastering the inner surface of a well at Rs 9.50 per m
2
, if it is 21 m deep and diameter of its top is 6 m.
S o l u t i o n :
Given: Height, h = 21 mDiameter, d = 6 mRadius, r = 3 mArea of the inner surface of the well, S = 2 prh = 2 p ×3 ×21 m
2
= 2 ×
22
7
×3 ×21 m
2
= 396 m
2
According to question, the cost per m
Q u e s t i o n : 1 2
A cylindrical vessel open at the top has diameter 20 cm and height 14 cm. Find the cost of tin-plating it on the inside at the rate of 50 paise per hundred square centimetre.
S o l u t i o n :
Given: Diameter, d = 20 cmRadius, r = 10 cmHeight, h = 14 cmArea inside the cylindrical vessel that is to be tin -plated = SS = 2 prh + pr
2
= 2 p ×10 ×14 + p×10
2
= 280 p +100 p = 380 ×
22
7
Q u e s t i o n : 1 3
The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find the cost of plastering its inner curved surface at Rs 4 per square metre.
S o l u t i o n :
Given: Inner diameter of the circular well = 3. 5 m ? Inner radius of the circular well, r = 1. 75 mDepth of the circular well, h = 10 mInner curved surface area, S = 2 prhS = 2 p ×1. 75 ×10 m
Q u e s t i o n : 1 4
The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions moving once over to level a playground. What is the area of the playground?
S o l u t i o n :
Given: Diameter of the roller = 84 cm ? Radius, r =
Diameter
2
= 42 cmIn 1 revolution, it covers the distance of its lateral surface area. Roller is a cylinder of height,  h = 120 cm  Radius = 42
Q u e s t i o n : 1 5
Twenty one cylindrical pillars of the Parliament House are to be cleaned. If the diameter of each pillar is 0.50 m and height is 4 m, what will be the cost of cleaning them at the rate of Rs
2.50 per square metre?
S o l u t i o n :
Given: Diameter of the pillars = 0. 5 mRadius of the pillars, r = 0. 25 mHeight of the pillars, h = 4 mNumber of pillars = 21Rate of cleaning = Rs 2. 50 per square metreCurved surface area
Q u e s t i o n : 1 6
The total surface area of a hollow cylinder which is open from both sides is 4620 sq. cm, area of base ring is 115.5 sq. cm and height 7 cm. Find the thickness of the cylinder.
S o l u t i o n :
Given: Total surface area of the cylinder = 4620 cm
2
Area of the base ring = 115. 5 cm
2
Height, h = 7 cmLet R be the radius of the outer ring and r be the radius of the inner ring. Area of the 
Q u e s t i o n : 1 7
The sum of the radius of the base and height of a solid cylinder is 37 m. If the total surface area of the solid cylinder is 1628 m
2
, find the circumference of its base.
S o l u t i o n :
Let r and h be the radius and height of the solid cylinder. Given: r+h = 37 mTotal surface area, S = 2 pr(r+h)1628 = 2 p ×r×37        r =
1628
2 p ×37
          =
1628
232.477
          = 7 mCircumference of its base
Q u e s t i o n : 1 8
Find the ratio between the total surface area of a cylinder to its curved surface area, given that its height and radius are 7.5 cm and 3.5 cm.
S o l u t i o n :
Let S
1
 and S
2
 be the total surface area and curved surface area, respectively. Given: Height, h = 7. 5 cmRadius, r = 3. 5 cmS
1
= 2 pr(r+h)S
2
= 2 prhAccording to the question:
S 1
S 2
=
2 p r(r+h)
2 p rh
S 1
S 2
=
Q u e s t i o n : 1 9
A cylindrical vessel, without lid, has to be tin-coated on its both sides. If the radius of the base is 70 cm and its height is 1.4 m, calculate the cost of tin-coating at the rate of Rs 3.50 per
1000 cm
2
.
S o l u t i o n :
Let r cm and h cm be the radius of the cylindrical vessel. Given: Radius, r = 70 cmHeight, h = 1. 4 m = 140 cmRate of tin -plating = Rs 3. 50 per 1000 square centimetreCost of tin -plating
Q u e s t i o n : 2 0
Find the volume of a cylinder whose
i
r = 3.5 cm, h = 40 cm
ii
r = 2.8 m, h = 15 m
S o l u t i o n :
i Given : r = 3. 5 cm, h = 40 cmVolume of cylinder, V = pr
2
h  =
22
7
×(3. 5)
2
×40  = 1540 cm
3
ii Given: r = 2. 8 m, h = 15 mVolume of cylinder, V = pr
2
h  =
22
7
×(2. 8)
2
×15  = 369. 6 m
3
Q u e s t i o n : 2 1
Find the volume of a cylinder, if the diameter (d) of its base and its altitude (h) are:
i
d = 21 cm, h = 10 cm
ii
d = 7 m, h = 24 m
S o l u t i o n :
i Given: d = 21 cm, radius, r =
d
2
= 10. 5 cmheight, h = 10 cmVolume of the cylinder, V = pr
2
h =
22
7
×(10. 5)
2
×10 = 3465 cm
3
ii Given: d = 7 m, radius, r =
d
2
= 3. 5 mheight h = 24 mVolume
Q u e s t i o n : 2 2
The area of the base of a right circular cylinder is 616 cm
2
 and its height is 25 cm. Find the volume of the cylinder.
S o l u t i o n :
Let the area of the base of a right circular cylinder be S cm
2
. Given: S = 616 cm
2
Height, h = 25 cmLet the radius of a right circular cylinder be r cm.    S = pr
2
616 =
22
7
×r
2
 r
   2
=
616×7
22
  r
2
 = 196
Q u e s t i o n : 2 3
The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find the volume of the cylinder.
S o l u t i o n :
Let r cm be the radius of a cylinder. Circumference of the cylinder, S = 2 prGiven: Height, h = 15 cmCircumference, S = 88 cm S = 2 pr88 = 2 ×
22
7
×r  r =
88×7
44
  r = 14 cmVolume of cylinder, V
Q u e s t i o n : 2 4
A hollow cylindrical pipe is 21 dm long. Its outer and inner diameters are 10 cm and 6 cm respectively. Find the volume of the copper used in making the pipe.
S o l u t i o n :
Let the length of the cylinder pipe be h = 21, dm = 210 cm. Let the outer and the inner radius of the pipe be R cm and r cm, respectively. ? 2R = 10  and 2r = 6R = 5 cm and r = 3cmVolume
Q u e s t i o n : 2 5
Find the i
( ) ( )
( ) ( )
curved surface area ii
total surface area and iii
volume of a right circular cylinder whose height is 15 cm and the radius of the base is 7 cm.
S o l u t i o n :
Given: Height, h = 15 cmRadius, r = 7 cm i Curved surface area, S
1
= 2 prh   = 2 ×
22
7
×7 ×15   = 660 cm
2
ii Total surface area, S
2
= 2 pr(r+h)   = 2 ×
22
7
×7 ×(7 +15)   = 44 ×22   = 968
Q u e s t i o n : 2 6
The diameter of the base of a right circular cylinder is 42 cm and its height is 10 cm. Find the volume of the cylinder.
S o l u t i o n :
Given: Diameter, d = 42 cmRadius, r =
d
2
= 21 cmHeight, h = 10 cmVolume of the cylinder, V = pr
2
h  =
22
7
×21
2
×10  = 13860 cm
3
Q u e s t i o n : 2 7
Find the volume of a cylinder, the diameter of whose base is 7 cm and height being 60 cm. Also, find the capacity of the cylinder in litres.
S o l u t i o n :
Given: Diameter, d = 7 cmRadius, r = 3. 5 cmHeight, h = 60 cmVolume of the cylinder, V = pr
2
h  =
22
7
×3. 5
2
×60  = 2310 cm
3
Capacity of the cylinder in litres =
2310
1000
           ( 1litre = 1000
Q u e s t i o n : 2 8
A rectangular strip 25 cm × 7 cm is rotated about the longer side. Find the volume of the solid, thus generated.
S o l u t i o n :
Given: Rectangular strip has radius, r = 7 cmHeight, h = 25 cmVolume of the solid, V = pr
2
h  =
22
7
×7
2
×25  = 3850 cm
3
Q u e s t i o n : 2 9
A rectangular sheet of paper, 44 cm × 20 cm, is rolled along its length to form a cylinder. Find the volume of the cylinder so formed.
S o l u t i o n :
The length l and breadth b of the rectangular sheet are 44 cm and 20 cmNow, the sheet is rolled along the length to form a cylinder. Let the radius of the cylinder be r cm. Height, h =
Q u e s t i o n : 3 0
The volume and the curved surface area of a cylinder are 1650 cm
3
 and 660 cm
2
 respectively. Find the radius and height of the cylinder.
S o l u t i o n :
Curved surface area of the cylinder = 2 prh  =660 cm
2
     ... 1
Volume of the cylinder = pr
2
h   =1650 cm
3
                       ... 2
From 1
and 2
, we can calculate the radius (r) and the height of cylinder (h).
We know the volume of the cylinder, i.e. 1650 cm
3
?  1650 = pr
2
h
     
   h = 
1650
p r
2
Substituting h into 1
:
660 = 2 prh
660 = 2 pr × 
1650
p r
2
660r = 21650
r = 5 cm
h = 
1650
p r
2
     = 
1650
22
7
×5
2
 = 21 cm.
Hence, the radius and the height of the cylinder are 5 cm and 21 cm, respectively.
Q u e s t i o n : 3 1
The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their volumes.
S o l u t i o n :
Here, r
1
 = Radius of cylinder 1
         h
1
 = Height of cylinder 1
         r
2
 = Radius of cylinder 2
         h
2
 = Height of cylinder 2
         V
1
 = Volume of cylinder 1
         V
2
 = Volume of cylinder 2
Ratio of the radii of two cylinders = 2:3
Ratio of the heights of two cylinders = 5:3
Volume of the cylinder =  pr
2
h
V
1
/V
2
 = ( pr
1
2
h
1
)/( pr
2
2
h
2
) = ( p(2r)
2
5h)/( p(3r)
2
3h) 
V
1
/V
2
 = ( p4r
2
5h)/( p9r
2
3h) = 20 / 27
Hence, the ratio of their volumes is 20:27
Q u e s t i o n : 3 2
( ) ( )
( ) ( )
The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. Find the volume of the cylinder, if its total surface area is 616 cm
2
.
S o l u t i o n :
Let r cm be the radius and h cm be the length of the cylinder. The curved surface area and the total surface area is 1:2.
The total surface area is 616 cm
2
.
The curved surface area is the half of 616 cm
2
, i.e. 308 cm
2
?.
Curved area = 2 prh
       So, h = 
308
2 p r
Total surface area = Curved surface area + Top and bottom area
Top and bottom area = 616 - 308 = 308 cm
2
 = 2 pr
2
r
2
 = 
308
2×
22
7
r = 7 cm
h = 
308
2× p ×7
= 7cm
Then, the volume of the cylinder can be calculated as follows:
V = 
22
7
× 7
2
 × 7 = 1078
Hence, it is obtained that the volume of the cylinder is 1078 cm
3
.
Q u e s t i o n : 3 3
The curved surface area of a cylinder is 1320 cm
2
 and its base has diameter 21 cm. Find the volume of the cylinder.
S o l u t i o n :
r cm = Radius of the cylinder
h cm = Height of the cylinder
Diameter of the cylinder is 21 cm. Thus, the radius is 10.5 cm.
Since the curved surface area has been known, we can calculate h by the equation given below:
The curved surface area of the cylinder = 2 prh
1320 cm
2
= 2 prh
1320 cm
2
= 2 x 22 x 10.5cm
x h
?                       7
h = 20 cm
? Volume of the cylinder (V) = pr
2
h
V = 2210.5cm
2
20cm
        7
V = 6930 cm
3
Q u e s t i o n : 3 4
The ratio between the radius of the base and the height of a cylinder is 2 : 3. Find the total surface area of the cylinder, if its volume is 1617 cm
3
.
S o l u t i o n :
Let r cm be the radius and h cm be the height of the cylinder. It is given that the ratio of r and h is 2:3, so h = 1.5r
The volume of the cylinder (V) is 1617 cm
3
.
So, we can find the radius and the height of the cylinder from the equation given below:
V= pr
2
h
1617 = pr
2
h
1617 = pr
2
(1.5r)
 r
3
 =343
r = 7 cm and h = 10.5 cm
Total surface area = 2 pr
2
+2 prh
?                                =2 ×
22
7
×7
2
+2 ×
22
7
×7 ×10. 5 = 770 cm
2
                              
Hence, the total surface area of the cylinder is 770 cm
2
.
Q u e s t i o n : 3 5
The curved surface area of a cylindrical pillar is 264 m
2
 and its volume is 924 m
3
. Find the diameter and the height of the pillar.
S o l u t i o n :
Here, r  m= radius of the cylinder
         h m= height of the cylinder
Curved surface area of the cylinder = 2 prh     ... 1
Volume of the cylinder = pr
2
h                        ... 2
            924 = pr
2
h
            h =
924
p r
2
Then, substitute h into equation 1
:
264 = 2 prh
264 = 2 pr
924
p r
2
264r = 2924
      r =
2×924
264
r = 7 m, so d = 14 m
h =
924
p r
2
 h=
924
22
7
×7
2
= 6 m
Hence, the diameter and the height of the cylinder are 14 m and 6 m, respectively.
Q u e s t i o n : 3 6
Two circular cylinders of equal volumes have their heights in the ratio 1 : 2. Find the ratio of their radii.
S o l u t i o n :
Here, V
1
 = Volume of cylinder 1
         V
2
 = Volume of cylinder 2
          r
1
 = Radius of cylinder 1
          r
2
 = Radius of cylinder 2
          h
1
 = Height of cylinder 1
          h
2
 = Height of cylinder 2
Volumes of cylinder 1 and 2 are equal.
Height of cylinder 1 is half the height of cylinder 2.
? V
1 
= V
2
( pr
1
2
h
1
) = ( pr
2
2
h
2
) 
( pr
1
2
h) = ( pr
2
2
2h) 
r 1
2
r 2
2
=
2
1
r 1
r 2
=
2
1
Thus, the ratio of their radii is v 2
 : 1.
Q u e s t i o n : 3 7
The height of a right circular cylinder is 10.5 m. Three times the sum of the areas of its two circular faces is twice the area of the curved surface. Find the volume of the cylinder.
S o l u t i o n :
It is known that three times the sum of the areas of the two circular faces, of the right circular cylinder, is twice the area of the curved surface.
Hence, it can be written using the following formula:
3 (2 pr
2
) = 2(2 prh)
3 pr
2
 = 2 prh
3r = 2h
It is known that the height of the cylinder (h) is 10.5 m.
Substituting this number in the equation:
3r = 210.5
r = 7 m
Volume of the cylinder = pr
2
h
                             = 22 (7
2
) 10.5
                                7
                             = 1617 m
3
Thus, the volume of the cylinder is 1617 m
3
.
Q u e s t i o n : 3 8
How many cubic metres of earth must be dug-out to sink a well 21 m deep and 6 m diameter?
S o l u t i o n :
The volume of the earth that must be dug out is similar to the volume of the cylinder which is equal to pr
2
h.
Height of the well =21 m
Diameter of the well= 6 m
? Volume of the earth that must be dug out = ( p (3
2
) 21
) m
3
= 594 m
3
Q u e s t i o n : 3 9
The trunk of a tree is cylindrical and its circumference is 176 cm. If the length of the trunk is 3 m, find the volume of the timber that can be obtained from the trunk.
S o l u t i o n :
Circumference of the tree = 176 cm = 2 pr
Length of the trunk, h= 3 m =300 cm
So, the radius (r) can be calculated by:
r =
176
2×
22
7
= 28 cm
( )
v