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## Mathematics Surface Area And Volume Worksheets for Class 10

We have provided **chapter-wise worksheets for class 10 Mathematics Surface Area And Volume** which the students can download in Pdf format for free. This is the best collection of Mathematics Surface Area And Volume standard 10th worksheets with important questions and answers for each grade 10th Mathematics Surface Area And Volume chapter so that the students are able to properly practice and gain more marks in Class 10 Mathematics Surface Area And Volume class tests and exams.

### Chapter-wise Class 10 Mathematics Surface Area And Volume Worksheets Pdf Download

**Question. The volume of the greatest sphere that can be cut off from a cylindrical log of wood of base radius 1 cm and height 5 cm is **

(A) 4/3 π

(B) 10/3π

(C) 5π

(D) (20/3)5π

**Answer**

A

**Question. The surface area of a sphere is same as the curved surface area of a right circular cylinder whose height and diameter are 12 cm each. The radius of the sphere is **

(A) 3 cm

(B) 4 cm

(C) 6 cm

(D) 12 cm

**Answer**

D

**Question. The ratio of the total surface area to the lateral surface area of a cylinder with base radius 80 cm and height 20 cm is **

(A) 2:1

(B) 3:1

(C) 4:1

(D) 5:1

**Answer**

D

**Question. The dimensions of a cuboid are in the ratio of 1 : 2 : 3 and its total surface area is 88 m2. The dimensions are **

(A) 2m, 4m, 6m

(B)1 m, 2 m, 3 m

(C) 4 m, 5 m, 6 m

(D) 6m, 8m, 10 m

**Answer**

A

**Question. The curved surface area of one cone is twice that of the other cone. The slant height of the second is twice that of the first one, then ratio of their radii is **

(A) 4 : 1

(B) 3 : 1

(C)2 : 1

(D)5 : 1

**Answer**

A

**Question. On increasing each of the radius of the base and the height of a cone by 20%, then its volume will be increased by **

(A) 20%

(B) 40%

(C) 60%

(D) 72.8%

**Answer**

D

**Question. The volumes of two spheres are in the ratio 64 : 27. The ratio of their surface area is **

(A)9 : 16

(B) 16 : 9

(C)3 : 4

(D)4 : 3

**Answer**

B

**Question. In a shower, 5 cm of a rain falls. The volume of the water that falls on 2 hectares of ground, is **

(A) 100 m^{3}

(B) 10 m^{3}

(C) 1000 m^{3}

(D) 10000 m^{3}

**Answer**

C

**Question. The sum of length, breadth and height of cuboid is 19 cm and its diagonal is 5 5cm . Its surface area is **

(A) 361 cm^{2}

(B) 125 cm^{2}

(C) 236 cm^{2}

(D) 486 cm^{2}

**Answer**

C

**Question. Volumes of two solid spheres are in the ratio 125 : 64. Determine their radii, if the sum of their radii is 45 cm. **

(A) 25cm, 20cm

(B)15cm, 30cm

(C) 35cm, 10cm

(D) 40cm, 5cm

**Answer**

A

**Question. A semi-circular thin sheet of paper of diameter 28 cm is bent and an open conical cup is made. **

Find the capacity of the cone.

(A) 311.2 cm^{3}

(B) 622.36 cm^{3}

(C) 30.51 cm^{3}

(D)152 m^{3}

**Answer**

B

**Question. A solid sphere of radius x cm is melted and cast into a shape of a solid cone of same radius. Then the height of the cone is **

(A) 3x cm

(B) x cm

(C) 4x cm

(D) 2x cm

**Answer**

C

**Question. The volume of a wall which is 5 times as high as it is broad and 8 times as long as it is high, is 12.8 m3. The breadth of the wall is **

(A) 30 cm

(B) 40 cm

(C) 22.5 cm

(D) 25 cm

**Answer**

B

**Question. A mason constructs a wall of dimension (270 cm ×300 cm ×350 cm) with bricks, each of size (22.5 cm ×11.25 cm ×8.75 cm) and it is assumed that 1/8 space is covered by the cement. Number of bricks used to construct the wall is **

(A) 11000

(B) 11100

(C) 11200

(D) 11300

**Answer**

C

**Question. If the radii of the circular ends of frustum of a cone are 20 cm and 12 cm and its height is 6 cm, then the slant height of frustum (in cm) is **

(A) 10

(B) 8

(C) 12

(D) 15

**Answer**

A

**VERY SHORT ANSWER TYPE QUESTIONS**

**Question. If the volume and surface area of a sphere are numerically equal, then what is its diameter?**

**Answer**

6 units

**Question. What is the length of a diagonal of a cube that can be inscribed in a sphere of radius 6 cm?**

**Answer**

12 cm

**Question. What is the length of the largest rod that can be put in a box of inner dimensions 18 cm × 24 cm × 30 cm?**

**Answer**

30 √2 cm

**Question. If the semi-vertical angle of a cone of height 3 cm is 60°, find its volume.**

**Answer**

27π cm^{3}

**Question. What is the surface area of a cube, whose volume is 27 cm ^{3 }?**

**Answer**

54 cm^{2}

**Question. If a sphere of radius 6 cm is melted and drawn into a wire of diameter 1/5 cm, find the length of the wire.**

**Answer**

288 m

**Question. What is the length of the diagonal of a cube that can be inscribed in a sphere of radius 10 cm?**.

**Answer**

20 cm

**Question. The largest sphere is carved out of a cube of side 7 cm. What is the volume of the sphere in terms of π.**

**Answer**

343π cm3/6

**Question. Three cubes, whose edges are 6 cm, 8 cm and 10 cm are melted and formed into a single cube. What is the length of diagonal of the single cube?**

**Answer**

12 √3 cm

**Question. What is the radius of a solid hemisphere whose volume and the whole surface area are numerically equal?**

**Answer**

4.5 units

**Question. What is the height of a cone whose base area and volume are numerically equal?**

**Answer**

3 cm

**Question. If the surface area of a sphere is 616 cm2, what is its volume?**

**Answer**

1437.33 cm^{3}

**Question. The radius of a solid hemispherical toy is 3.5 cm. What is its total surface area?**

**Answer**

115.5 cm^{2}

**Question. Two cubes each of volume 64 cm3 are joined end to end. What is the surface area of the resulting cuboid?**

**Answer**

160 cm^{2}

**Question. If height of a frustum is 4 cm and the radii of two bases 3 cm and 6 cm respectively, what is the slant height of the frustum?**

**Answer**

5 cm

**Question. What is the slant height of a cone, if its height is 12 cm and radius is 5 cm?**

**Answer**

13 cm

**Question. What is the radius of a cylinder whose volume and curved surface area are numerically equal?**

**Answer**

2 units

**Question. What is the ratio of the volume of a cube to that of a sphere which will exactly fit inside the cube?**

**Answer**

6 : π

**Question. A rectangular sheet of paper 66 cm × 18 cm is rolled along its length and a cylinder is formed. What is curved surface area of the cylinder.**

**Answer**

1188 cm^{2}

**Question. The radii and heights of a cone, a hemisphere and cylinder are same. What is the ratio of the volumes of them.**

**Answer**

343 cm^{3}

**Short / Long Answer Type Questions :**

**Question. Two solid cones A and B are placed in a cylindrical tube as shown in the figure. The ratio of their capacities is 2 : 1. Find the heights and capacities of cones. Also, find the volume of the remaining portion of the cylinder. **

**Solution.** As the ratio of volumes of cones A and B is 2 : 1, their

radii are same equal to r = 6/2 = 3 cm .

**Question. A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The height and the radius of the cylindrical part are 13 cm and 5 cm respectively. The radii of the hemispherical and conical parts are the same as that of the cylindrical part. Calculate the surface area of the toy if height of the conical part is 12 cm.****Solution.** 770 cm^{2}

**Question. From a solid cylinder of height 12 cm and diameter of the base 10 cm a conical cavity of the same height and same diameter is hollowed out. Find the surface area of the remaining solid.****Solution.** Surface area of remaining solid

= 2π × 5 × 12 + π × 5 × 5 + π × 5 × √5^{2} + 12^{2}

= 120π + 25π + 65π

= 210π = 210 × 22/7 = 660 cm^{2}

**Question. A toy is in the form of a cone mounted on a hemisphere of radius 3.5 cm. The total height of the toy is 15.5 cm. Find the total surface area of the toy. ****Solution.** Here, r = 3.5 cm

∴ h = (15.5 – 3.5) cm

= 12.0 cm

Surface area of the conical part = πrl

**Question. A solid right circular cone of height 120 cm and radius 60 cm is placed in a right circular cylinder full of water of height 180 cm such that it touches the bottom. Find the volume of water left in cylinder, if the radius of the cylinder is equal to the radius to the cone.****Solution.** The water left in cylinder

= Volume of cylinder – Volume of cone

Volume of water left after immersing the cone into cylinder

full of water

= Volume of cylinder – Volume of cone

**Question. A toy is in the form of a cone mounted on a hemisphere with the same radius. The diameter of the base of the conical portion is 6 cm and its height is 4 cm. Determine the surface area of the toy. [Take π = 3.14]****Solution.** Surface area of toy

= 2π × 3 × 3 +π × 3 ×√32 + 42

= 18π + 15π = 33π = 33 × 22/7

= 103.62 cm^{2}

**Question. A solid is in the shape of a cone surmounted on a hemisphere. The radius of each of them being 3.5 cm and the total height of the solid is 9.5 cm. Find the volume of the solid.****Solution.** Let the radius of hemisphere or a cone, r = 3.5 cm

Also, the height of the cone,

h = 9.5 – 3.5 = 6 cm (2)

Volume of solid

= Volume of hemisphere + volume of cone

**Question. A solid is in the form of a right circular cylinder with hemispherical ends. The total height of the solid is 58 cm and the diameter of the cylinder is 28 cm. Find the total surface area of the solid. [Use π = 22/7]****Solution.** Total surface area of solid = Curved surface area of cylindrical portion + 2(Curved surface area of hemispherical portion)

= 2 × 22/7 x 14 x 30 + 2 (2 x 22/7 x 14 x 14)

= 2640 + 2464 = 5104 cm^{2}

**Question. A gulab jamun when completely ready for eating contains sugar syrup upto about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns shaped like a cylinder with two hemispherical ends, if the complete length of each of the gulab jamuns is 5 cm and its diameter is 2.8 cm. ****Solution.** 337.88 cm^{3}

**Question. A rocket is in the form of a right circular cylinder closed at the lower end and surmounted by a cone with the same radius as that of cylinder. The diameter and height of cylinder are 6 cm and 12 cm, respectively. If the slant height of the conical portion is 5 cm, then find the total surface area and volume of rocket. (Use π = 3.14)****Solution.** Volume of rocket

= Volume of cylinder + Volume of cone

**Question. The following figure shows a model which is the combination of a cone and cylinder. The total length of the model is 52 cm, the length of conical portion is 12 cm, the base of the conical portion has diameter of 10 cm and the base of the cylindrical portion is of diameter 6 cm. If the conical portion is to be painted orange and the cylindrical portion yellow, find the area of the model painted with each of these colours. **

**Solution.** orange: 254.34 cm^{2}, yellow: 781.86 cm^{2}

Hint: For conical portion, h = 12 cm,

**Question. What is the capacity of cylindrical vessel with the hemispherical bottom portion raised upwards? **

**Solution.** πr^{2}/3 [3h – 2r]

**Question. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown: If the height of the cylinder is 10 cm and its base is of radius 3.5 cm, find the total surface area of the article. **

**Solution.** Radius of the cylinder (r) = 3.5 cm

Height of the cylinder (h) = 10 cm

∴ Total surface area

**Question. A solid is hemispherical at the bottom and conical (of same radius) above it. If the surface areas of two parts are equal, then what is the ratio of its radius and the slant height of the conical part?****Solution.** 1 : 2

**Question. In the given figure, a decorative block is shown which is made of two solids, a cube and a hemisphere. The base of the block is a cube with edge 6 cm and the hemisphere fixed on the top has a diameter of 4.2 cm. Find [Take π = 22/7]****(a) the total surface area of the block.****(b) the volume of the block formed****Solution.** Diameter of the hemisphere = 4.2 cm

∴ Radius of the hemisphere = 2.1 cm

Edge of the cube (l) = 6 cm (1)

(a) Total surface area of the block

= Total surface area of the cube

– Base area of hemisphere

+ Curved surface area of hemisphere

**Question. A juice seller was serving his customers using glasses. The inner diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of the glass was 10 cm, find its actual capacity and its apparent capacity. (Use π = 3.14)****Solution.** Since the inner diameter of the glass = 5 cm and height

= 10 cm, the apparent capacity of the glass = πr^{2}h

= 3.14 × 2.5 × 2.5 × 10 cm^{3} = 196.25 cm^{3}

But the actual capacity of the glass is less by the volume of the hemisphere at the base of the glass.

**PRACTICE EXCERCISE**

**Question. A heap of wheat is in the form of a cone of radius 4.5 m and height 3.5 m. How much canvas cloth is required to just cover the heap?****Solution.** 80.61 m^{2}

**Question. The interior of a building is in the form of a cylinder of diameter 4.3 m height 3.8 m, surmounted by a cone whose vertical angle is a right angle. Find the area of the surface and the volume of the building.****Solution.** 71.828 m^{2}, 65.62 m^{3} (approx)

**Question. A hemispherical tank of radius 2.1 m is full of water. It is connected with a pipe which empties it at the rate of 7 litres per second. How much time will it take to empty the tank completely? (1 litre = 1000 cm ^{3})**

**Solution.**46.2 minutes

**Question. A cylindrical jar of radius 6 cm contains oil. Iron spheres each of radius 1.5 cm are immersed in the oil. How many spheres are necessary to raise the level of oil by 2 cm?****Solution.** 16

**Question. A barrel of fountain pen, cylindrical in shape, is 7 cm long and 5 mm in radius. A full barrel of ink is used for writing 165 words on an average. How many words would use a bottle of ink containing 1/10th of a litre.****Solution.** 3000 words

**Question. A cylindrical container has diameter 28 cm and contains sufficient water in it to submerge a cuboid 11 cm × 7 cm × 16 cm. Find the rise in level of the water when the cuboid is submerged in it.****Solution.** 2 cm

**Question. A semi-circular thin sheet of metal of diameter 28 cm, is bent to make an open conical cap. Find the capacity of the cap.****Solution.** 622.38 cm^{3}

**Question. A shuttlecock used for playing badminton has the shape of a frustum of a cone mounted on a hemisphere. The external diameters of the frustum are 5 cm and 2 cm, and the height of the entire shuttlecock is 7 cm. Find its external surface area.****Solution.** 74.26 cm^{2}

**Question. If a cone of radius 10 cm is divided into two parts by drawing a plane through the mid-point of its axis, parallel to its base. Compare the volumes of the two parts. ****Solution.** 1 : 7

**Question. The radii of the internal and external surfaces of a metallic spherical shell are 3 cm and 5 cm respectively. It is melted and recast into a solid right circular cylinder of height 10,2/3 cm. Find the radius of the base of the cylinder.****Solution.** 3.5 cm

**Question. A bucket of height 8 cm and made up of copper sheet is in the form of a frustum of a right circular cone with radii of its lower and upper ends as 3 cm and 9 cm respectively. Calculate :****(i) the height of the cone of which the bucket is a part.****(ii) the volume of water which can be filled in the bucket.****(iii) the area of copper sheet required to make the bucket.****(Leave the answer in terms of π) ****Solution.** (i) 12 cm (ii) 312 π cm^{3} (iii) 129 π cm^{2}

**Question. The largest sphere is carved out of a cube of a side 7 cm. find the volume of the sphere.****Solution.** 179.66 cm^{3}

**Question. Determine the ratio of the volume of a cube to that of a sphere which will exactly fit inside the cube.****Solution.** 6 : π

**Question. A sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?****Solution.** 1 cm

**Question. A solid sphere of radius 3 cm is melted and then recast into small spherical balls each of diameter 0.6 cm. Find the number of balls thus outained.****Solution.** 1000

**Question. A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of the two of the balls are 1.5 cm and 2 cm respectively. Determine the radius of the third ball.****Solution.** 2.5 cm

**Question. How many spherical lead shots each 4.2 cm in diameter can be obtained from a rectangular solid of lead with dimensions 66 cm × 42 cm × 21 cm. [use π = 22/7]****Solution.** 1500

**Question. A hemispherical bowl of internal diameter 36 cm contains a liquid. This liquid is to be filled in cylindrical bottles of radius 3 cm and height 6 cm. How many bottles are required to empty the bowl?****Solution.** 72

**Question. A glass cylinder with diameter 20 cm has water to the height of 9 cm. A metal cube of 8 cm edge is immersed in it completely. Find the height by which the water will rise in the cylinder. [use π = 3.142[****Solution.** 1.6 cm

**Question. A conical vessel whose internal radius is 5 cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel with internal radius 10 cm. Find the height to which the water rises.****Solution.** 2 cm

**Question. The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 1/27 of the volume of the given cone, at what height above the base is the section made?****Solution.** 20 cm

**Question. A solid sphere of radius 6 cm is melted and recast into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is 4 cm and if its height is 76.8 cm, find the uniform thickness of the cylinder.****Solution.** 0.5 cm

**Question. An iron pillar has some part in the form of a right circular cylinder and the remaining in the form of a right circular cone. The radius of the base of each of the cone and the cylinder is 8 cm, the cylindrical part is 240 cm high and conical part is 36 cm high. Find the weight of the pillar, If 1 cm3. of iron weighs 10 grams. [use π = 22/7]****Solution.** 506.88 kg

**Question. Water flows at the rate of 10 metres per minute from a cylindrical pipe 5 mm in diameter. How long would it take to fill a conical vessel whose diameter at the surface is 40 cm and depth 24 cm?****Solution.** 51 minutes 12 seconds

**Question. The diameter of a sphere is 6cm. it is melted and drawn into a wire of diameter 2mm. Find the length of the wire.****Solution.** 3600cm or 36m

**Question. Metallic sphere of radii 6cm, 8cm and10cm respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.****Solution.** 12cm

**Question. A solid toy is in the form of a right circular cylinder with a hemispherical shape at one end and a cone at the other end. Their common diameter is 4.2cm and the height of the cylindrical and conical position is12cm and 7cm respectively. Find the volume of the solid toy.****Solution.** 218.064cm³

**Question. A hemispherical depression is cut from one face of the cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.****Solution.**l^{2}/4 (24+π)

**Question. The largest sphere is curved out of a cube of a side 7cm. Find the volume of the sphere.****Solution.** 179.67cm³

**Question. A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of π .****Solution.** π

**Question. A juice seller was serving his customers using glasses. The inner diameter of the cylindrical glass was 5cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of glass was 10cm, find what the apparent capacity of the glass was and what the actual capacity was.(use π = 3.14)****Solution.** 163.54cm³

**Question. The height of a cone is 30cm. A small cone is cut off at the top by a plane parallel to the base of its volume be 1/27 of the volume of the given cone, at what height above the base is the section made?****Solution.** 20 cm

**Question. The Surface Area of a Sphere is 616cm ^{2}. Find its radius.**

**Solution.**7 cm

**Question. The diameters of the ends of a frustum of a cone are32cm and 20cm. If its slant height is 10cm. Find the lateral surface area.****Solution.** 816.4cm²

**Question. If the radii of the circular ends of a conical bucket which is 45cm high are 28cm and 7cm. Find the capacity of the bucket.****Solution.** 48510 cm^{3}

**Question. The slant height of the frustum of a cone is 5cm. if the difference between the radii of its two circular ends is 4cm, write height of the frustum.****Solution.** 3cm

**Question. Two cones have their heights in the ratio1:3 and radii 3:1. What is the ratio of their volumes?****Solution.** 3:1

**Question. The radii of two cones are in the ratio 2:1 and their volumes are equal. What is the ratio their heights?****Solution.** 1:4

**Question. An oil funnel of tin sheet consists of a cylindrical portion 10cm long attached to a frustum of a cone. If the total height be 22cm, diameter of the cylindrical portion be 8cm and the diameter of the top of the funnel be18cm. Find the area of the tin required to make the funnel.****Solution.** 249π

**Question. A 20m deep well with diameter7misdugandtheearth from digging evenly spread out to form a platform 22m by14m. Find the height of the platform.****Solution.** 2.5 m

**Question. The height of a solid cylinder is 15 cm and diameter 7 cm. Two equal conical holes of radius 3 cm and height 4 cm are cut off. Find the volume and surface area of the remaining solid. **

**Solution.** 502.07 cm^{3}, 444.7 cm^{2}

**Question. A cylindrical can of internal diameter 21 cm contains water. A solid sphere whose diameter is 10.5 cm is lowered into the cylindrical can. The sphere is completely immersed in water. Find the rise in the water level, assuming that no water overflows.****Solution.** 1.75 cm

**Question. A right circular cone is divided by a plane parallel to its base in two equal volumes. In what ratio will the plane divide the axis of the cone?****Solution.** 1 : 2^{1/3} – 1

**Question. The radii of the circular ends of a solid frustum of a cone are 33 cm and 27 cm and its slant height is 10 cm. Find its total surface area. ****Solution.** 7599.42 cm^{2}

**Question. A spherical copper ball of a diameter 9 cm is melted and drawn into a wire of diameter 2 cm. Find the length of the wire in metres.****Solution.** 1.215 m

**Question. A bucket made of aluminium sheet is of height 20 cm and its upper and lower ends are of radius 25 cm and 10 cm respectively. Find the cost of making the bucket, if the aluminium sheet costs Rs. 70 per 100 cm ^{2}. [use π = 3.142]**

**Solution.**Rs. 2143.05

**Question. Water is flowing at the rate of 0.7 metres per second through a circular pipe whose internal diameter is 2 cm into a cylindrical tank, the radius of whose base is 40 cm. Determine the increase in the water level in 1/2 hour.****Solution.** 78.75 cm

**Question. Lead spheres of diameter 6 cm are dropped into beaker containing some water and are fully submerged. The diameter of the beaker is 18 cm. Find how many lead spheres have been dropped in it if the water rises by 40 cm?****Solution.** 90

**Question. A cylindrical jar is 25 cm high with internal diameter 7 cm. A metal cube of edge 4 cm is immersed in the jar. Find the rise in the level of water. [use π = 22/7]****Solution.** 1, 51/77 cm

**Question. A hemispherical bowl of internal radius 9 cm is full of liquid. This liquid is to be filled into cylindrical shaped small bottled each of diameter 3 cm and height 4 cm, How many bottles are necessary to empty the bowl? ****Solution.** 54

**Question. A cylindrical tub of radius 12 cm contains water to a depth of 20 cm. A spherical ball is dropped into the tub and the level of the water is raised by 6.75 cm. Find the diameter of the ball.****Solution.** 18 cm

**Question. An iron sphere of diameter 12 cm is dropped into a cylindrical can of diameter 24 cm containing water. Find the rise in the level of water when the sphere is completely immersed.****Solution.** 2cm

**Question. A solid cylinder of diameter 12 cm and height 15 cm is melted and recast into 12 toys in the shape of a right circular cone mounted on a hemisphere. Find the radius of the hemisphere and the total height of the toy if height of the cone is three times the radius.****Solution. **3 cm, 12 cm

**Question. A metallic sphere of radius 10.5 cm is melted and then recast into small cones , each of radius 3.5 cm and height 3 cm. Find how many cones are obtained? ****Solution.** 126

**Question. The diameters of the internal and external surfaces of a hollow spherical shell are 6 cm and 10 cm respectively. If it is melted and recast into a solid cylinder of diameter 14 cm, find the height of the cylinder. ****Solution.** 8/3 cm

**Question. The radii of the circular ends of a frustum of height 6 cm are 14 cm and 6 cm respectively. Find the lateral surface area and total surface area of the frustum.****Solution.** L.S.A = 628.57 cm^{2}, T.S.A. = 1357.71 cm^{2}

**Question. Four right circular cylindrical vessels each having diameter 21 cm and height 38 cm are full of ice-cream. The ice-cream is to be filled in cones of height 12 cm and diameter 7 cm having a hemispherical shape at the top. Find the total number of such cones which can be filled with ice-cream. ****Solution.** 216

**Question. A wooden toy rocket is in the shape of a cone mounted on a cylinder, as shown in the given figure. The height of the entire rocket is 26 cm, while the height of the conical part is 6 cm. The base of the conical portion has a diameter of 5 cm, while the base diameter of the cylindrical portion is 3 cm. If the conical portion is to be painted orange and the cylindrical portion yellow, find the area of the rocket painted with each of these colours. (use π = 3.14) **

**Solution.** Area to be painted orange = 63.585 cm^{2}, Area to be painted yellow = 195.465 cm^{2}

**Question. A milk container is made of metal sheet in the form of a frustum of a cone and is of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk which the container can hold when fully filled at Rs 16 per litre and, the cost of the metal sheet used in making the container, at Rs 5 per sq. dm. [use π = 3.142]****Solution.** cost of milk = Rs. 167.20, cost of metal sheet used = Rs. 97.97

**Question. A well, whose diameter is 7 m, has been dug 22.5 m deep and the earth dug out is used to form an embankment 10.5 m wide around it. Find the height of the embankment.****Solution.** 1.5 m

**Question. If the radii of the circular ends of a conical bucket which is 45 cm high be 28 cm and 7 cm, find the capacity of the bucket. ****Solution.** 48510 cm^{3}

**Question. A hollow cone is cut by a plane parallel to the base and the upper partion is removed.If the curved surface of the remainder is 8/9 of the curved surface of the whole cone, find the ratio of the line-segment into which the cones altitude is divided by the plane.****Solution.** 1 : 2

**Question. Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 9 cm.****Solution.** 190.93 cm^{3}

**Question. An agricultural field is in the form of a rectangle of length 20 m and width 14 m. A 10 m deep well of diameter 7 m is dug in a corner of the field and the earth taken out of the well is spread evenly over the remaining part of the field. Find the rise in its level.****Solution.** 1.6 m (approx)

**Question. A right triangle whose sides are 15 cm and 20 cm, is made to revolve above its hypotenuse. Find the volume and the surface area of the double cone so formed. (use π = 3.14).****Solution.** 3768 cm^{3} , 1318.8 cm^{2}

**Question. A cylindrical bucket 42 cm in diameter and 50 cm high is full of water. The water is emptied into a rectangular tank 66 cm long and 21 cm wide. Find the height of the water level in the tank.****Solution.** 50 cm

**Question. The base radius and height of a right circular solid cone are 2 cm and 8 cm respectively. It is melted and recast into spheres of diameter 2 cm each. Find the number of spheres so formed.****Solution.** 8

**Question. Water in a canal, 30 dm wide and 12 dm deep is flowing with velocity of 10 km/hr. How much area will it irrigate in 1/2 hour, if 8 cm of standing water is required for irrigation?****Solution.** 225000 m^{2}

**Question. Water is being pumped out through a circular pipe whose internal diameter is 7 cm. If the flow of water is 72 cm per second, how many litres of water are being pumped out in one hour?****Solution.** 9979.2 l

**Question. Water is flowing at the rate of 2.5 km/hr through a circular pipe of 20 cm internal diameter into a circular cistern of diameter 20 metres and depth 2.5 metres. In how much time will the cistern be filled?****Solution.** 10 hours

**Question. A rectangular vessel is 20 cm × 16 cm × 11 cm and is full of water. This water is poured into a conical vessel of base radius 10 cm. If the vessel is completely filled , find the height of the conical vessel.****Solution.** 33.6 cm

**Question. Find the volume and the surface area of a solid in the form of a right circular cylinder with hemispherical ends whose total length is 2.7 m and the diameter of each hemispherical ends is 0.7 m.****Solution.** 0.9496 m^{3}, 5.94 m^{2}

**Question. A friction clutch is in the form of frustum of a cone, the diameters of the ends being 32 cm and 20 cm and length 8 cm. Find its bearing surface and volume.****Solution.** L.S.A. = 817.14 cm^{2}, vol. =4324.57 cm^{3}

**Question. A bucket is in the form of a frustum of a cone with a capacity of 12308.8 cm ^{3} of water. The radii of the top and bottom circular ends are 20 cm and 12 cm respectively. Find the height of the bucket and the area of the metal sheet used in its making. [use π = 3.142]**

**Solution.**height = 15 cm, Area = 2160.32 cm

^{2}

**Question. A conical vessel of radius 6 cm and height 8 cm is completely filled with the water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed as shown in the given figure. What fraction of the water over flows? **

**Solution.** 3 : 8

**Question. The radii of the ends of a bucket 30 cm high are 21 cm and 7 cm. Find its capacity in litres and the amount of sheet required to make this bucket and its cost at Rs. 2 per sq. dm of sheet.****Solution.** capacity = 20.02 l, amount of sheet required = 3068.8 cm2, Total cost = Rs. 61.34

**Question. A hemispherical tank of radius 1, 3/4 m is full of water It is connected by a cylindrical pipe which empties it at 7 litres per second. Find the time it will take to empty the tank completely.****Solution.** 26.73 minutes

**Question. If the number of square centimetres on the surface of a sphere is equal to the number of cubic centimetres in its volume, what is the diameter of the sphere?****Solution.** 6 cm

**Question. A solid right circular cone of base radius 6 cm and height 12 cm is made up of metal. A right circular cone of height 4 cm is removed from the original cone leaving a frustum as shown. **

**Find :****(i) the radius of the right circular cone which has been removed.****(ii) the volume of the frustum.****(iii) the whole surface area of the frustum.****Solution.** (i) 2 cm (ii) 435.8 cm3 (iii) 350.58 cm^{2}

**Question. The diameter of a metallic solid sphere is 12 cm. It is melted and drawn into a wire having diameter of the cross-section 0.2 cm. Find the length of the wire.****Solution.** 288 m

**Question. A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 41, 19/21 cm ^{3} of air. If the internal diameter of the building is equal to the total height above the floor, find the height of the building.**

**Solution.**4 m

**Question. A tent is in the form of a cylinder of diameter 20 m and height 2.5 m, surmounted by a cone of equal base and height 7.5 m. Find the capacity of the tent and the cost of the canvas at Rs. 100 per square meter.****Solution.** 500 π m^{2}, Rs. 55000

**Question. A circus tent is in the form of a right circular cylinder and a right circular cone above it. The diameter tent is 21 m. Find the total surface area of the tent.****and the height of the cylindrical part of the tent are 126 m and 5 m respectively. The total height of the [use π = 22/7 ]****Solution.** 14850 m^{2}

**Question. A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 19 cm and the diameter of the cylinder is 7 cm. Find the volume and total surface area of the solid. [use π = 22/7 ] ****Solution.** 641.66 cm3, 418 cm^{2}

**Question. An ice-cream cone consists of a right circular cone of height 14 cm and diameter of the circular top is 5 cm. It has hemisphere on the top with the same diameter as of circular top. Find the volume of icecream in the cone.****Solution.** 124.4 cm^{3}

**Question. A solid is composed of a cylinder with hemispherical ends. It the whole length of the solid is 104 cm and the radius of each of the hemispherical ends is 7 cm, find the cost of polishing its surface at the rate of Rs. 10 per dm ^{2}. **

**Solution.**Rs. 457.60

**Question. A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The radius and height of the cylindrical part are 5 cm and 13 cm respectively. The radii of the hemispherical and conical parts are the same as that of the cylindrical parts. Find the surface area of the toy if the total height of the toy is 30 cm. ****Solution.** 770 cm^{2}

**Question. A cylindrical road roller made of iron is 1 m long. Its internal diameter is 54 cm and the thickness of the iron sheet used in making the roller is 9 cm. Find the mass of the roller, if 1 cm ^{3} of iron has 7.8 gm mass. [ use π = 3.14]**

**Solution.**1388.7 kg

**Question. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by a cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm3 of iron has approximately 8 g mass. [use π = 3. 14] ****Solution.** 892.26 kg

**Question. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent , Also find the cost of the canvas of the tent at the rate of Rs. 500 per m ^{2}. **

**Solution.**44 m

^{2}, Rs. 22000

**Question. Three cubes each of side 5 cm are joined end to end. Find the surface area of the resulting cuboid.****Solution.** 350 cm^{2}

**Question. A vessel is in the form of inverted cone. Its height is 8 cm and radius of its top, which is open , is 5 cm. It is filled with water upto the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped in the vessel, one fourth of the water flows out. Find the number of lead shots dropped in the vessel. ****Solution.** 100

**Question. A circus tent is cylindrical upto a height of 3 m and conical above it. If the diameter of the base is 105 m and the slant height of the conical part is 53 m, find the total canvas used in making the tent.****Solution.** 9735 m^{2}

**Question. A solid wooden toy is in the shape of a right circular cone mounted on a hemisphere. If the radius of the hemisphere is 4.2 cm and the total height of the toy is 10.2 cm, find the volume of the wooden toy.****Solution.** 266.11 cm^{3}

**Question. A cylindrical container of radius 6 cm and height 15 cm is filled with ice- cream. The whole ice-cream has to be distributed to 10 children in equal cones with hemispherical tops. If the height of the conical portion is four times the radius of its base, find the radius of the ice-cream cone.****Solution.** 3 cm

**Question. A cylindrical tub of radius 5 cm and length 9.8 cm is full of water. A solid in the form of a right circular cone mounted on a hemisphere is 3.5 cm and height of the cone outside the hemisphere is 5 cm, find the volume of the water left in the tub. [use π = 22/7 ]****Solution.** 616 cm^{3}

**Question. A solid is in the form of a right circular cone mounted on a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 4 cm. The solid is placed in cylindrical tub, full of water, in such a way that the whole solid is submerged in water. If the radius of the cylinder is 5 cm and its height is 10.5 cm, find the volume of water left in the cylindrical tub. [use π = 22/7 ]****Solution.** 683.83 cm^{3}

**Question. A solid is composed of a cylinder with hemispherical ends. If the whole length of the solid is 108 cm and the diameter of the hemispherical ends is 36 cm, find the cost of polishing the surface of the solid at the rate of 10 paisa per sq. cm. [use π = 22/7 ] ****Solution.** 1221.94 Rs.

**Question. The interior of a building is in the form of a right circular cylinder of diameter 4.2 m and height 4 m, surmounted by a cone. The vertical height of the cone is 2.1 m. Find the outer surface area and the volume of the building. [use π = 22/7 ]****Solution.** 72.35 m2, 65.15 m^{3}

**Question. A solid toy is in the form of a right circular cylinder with a hemispherical shape at one end and a cone at the other end. Their common diameter is 4.2 cm and the height of the cylindrical and conical portions are 12 cm and 7 cm respectively. Find the volume of the solid toy. [use π = 22/7 ] ****Solution.** 218.064 cm^{3}

**Question. A Gulab Jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found is 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see figure). ****Solution.**

**Question. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the surface area of the remaining solid to the nearest cm ^{2}. **

**Solution.**For a conical part, we have r = 14/20 cm 0.7 cm, h = 2.4 cm

**Question. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in the given figure. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article. ****Solution.** For a given solid, we have

r = 7/2 cm, h = 10 cm

Total surface area of the solid by scooping out a hemisphere from

each end of a cylinder

= CSA of the cylinder + 2 × (CSA of the hemisphere)

= 2πrh + 2 (2πr^{2})

= 2πr (h + 2r)

**Question. A container, opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm, respectively. Find the cost of the milk which can be completely fill the container, at the rate of Rs. 20 per litre. Also find the cost of metal sheet used to make the container, if it costs Rs. 8 per 100 cm2 [Take π = 3.14] ****Solution.** Here, h = 16 cm, r = 8 cm, R = 20 cm

**Question. A metallic right circular cone height 20 cm and whose vertical angle is 60° is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter 1/16 cm, find the length of the wire. ****Solution.** Clearly, OE = 1/2 OF = 1/2 x 20 cm 10 cm

**Question. A tent is of the shape of a right circular cylinder upto a height of 3 metres and then becomes a right circular cone with a maximum height of 13.5 metres above the ground. Calculate the cost of painting the inner side of the tent at the rate of Rs. 2 per square metre, if the radius of the base is 14 metres. ****Solution.** here, radius of the base (r) = 14 cm and height (h) of the cylinder = 3m.

CSA of the cylinder = 2 πrh

**Question. A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm. ****Solution.** For a conical part ; r = 60 cm, h = 120 cm.

For a hemisphere; r = 60 cm

and, for a cylinder r = 60 cm, H = 180 cm

Now, volume of the solid consisting of a cone over a hemisphere

= volume of cone + volume of hemisphere

**Question. A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment. ****Solution.** Radius of the well (r) = 3/2 m 1.5 m, and height (h) = 14 m

**Question. A vessel is in the form of a hemispherical bowl mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find its capacity. [Take π = 22/7 ] ****Solution.** For cylindrical part; r = 7 cm, h = 6 cm

and, for hemispherical part; r = 7 cm.

Total capacity of the bowl

= volume of the cylinder + volume of the hemisphere

**Question. A toy is in form of a cone mounted on a hemi-sphere of radius 3.5 cm. The total height of the toy is 15.5 cm. Find its total surface area. ****Solution.** Here, AO = AC – OC = 15.5 cm – 3.5 cm = 12 cm

**Question. A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of the wood in the entire stand. ****Solution.** For a cuboid, l = 15 cm, b = 10 cm, h = 3.5 cm

∴ Volume of cuboid = l b h

**Question. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid. ****Solution.** Here r = 1/2 , edge of a cube = l

Total surface area of the cube after hemispherical depression

= TSA of a cube – base area of a hemisphere + CSA of a hemisphere

= 6l^{2} – πr^{2} + 2πr^{2} = 6l^{2} + πr^{2}

**Question. A sphere, of diameter 12 cm, is dropped in a right circular cylindrical vessl, partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by 3 , 5/9 cm. Find the diameter of the cylindrical vessel. ****Solution.** Here, radius of a sphere (r) = 12/2 cm=6 cm

**Question. A container shaped like a right circular cylinder having diamater 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream. ****Solution.** For cylindrical container, we have R = 12/2 cm = 6 cm and H = 15 cm

Volume of the cylindrical icecream container = πR2 H

= 22/7 x 6 x 6 x 15 cm^{2}

**Question. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in her field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/ hr, in how much time will the tank be filled? ****Solution.** Speed of water = 3 km/hr = 3000 m/hr

**Question. A solid right circular cone of diameter 14 cm and height 8 cm is melted to form a hollow sphere. If the external diameter of the sphere is 10 cm, find the internal diameter of the sphere. ****Solution.** For cone; radius (r) = 14/2 cm 7 cm and height (h) = 8 cm

**Question. A circus tent has cylindrical shape surmounted by a conical roof. The radius of the cylindrical base is 20 m. The height of the cylindrical and conical portions are 4.2 m and 2.1 m respectively. ****Find the volume inside the tent.****Solution.** Let h1 = 4.2 m be the height of cylindrical portion, h_{2} = 2.1 m be the height of conical portion and r = 20 m be the radius of tent.

Then, the volume inside the tent

**Question. Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions** **of the model to be nearly the same.) ****Solution.** The figure shows the model as per the given question:

Now, for the cylinder

**Question. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find: ****(i) the curved surface area of the cavity formed.****(ii) the total surface area of the remaining solid to the nearest cm ^{2}.**

**Solution.**Since, diameter = 1.4 cm, radius = 1.4/2 cm = 0.7 cm

(i) If l is the slant height of the conical cavity hollowed out

l

^{2}= h

^{2}+ r

^{2}= (2.4)

^{2}+ (0.7)

^{2}= 5.76 + 0.49 = 6.25

**Question. A playing top (lattu) is shaped like a cone surmounted by a hemisphere. The entire top is 5 cm in height and the diameter of top is 3.5 cm. Find its whole surface area.****Solution. **

**Question. A pen stand (as shown in the figure) made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand. ****Solution.** Volume of the wood in the entire stand = Volume of cuboid – 4 × Volume of one conical depression

**Question. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass** **of the pole, given that 1 cm ^{3} of iron has approximately 8 g mass. [Use π = 3.14] 11**

**Solution.**Volume of the pole

= Volume of the bigger cylinder + Volume of the smaller cylinder

= πR

^{2}H + πr

^{2}h

= 3.14 × (12)

^{2}× 220 cm

^{3}+ 3.14 × (8)

^{2}× 60 cm

^{3}

= 111532.8 cm

^{3}

Since, 1 cm

^{3}of iron has approximately 8 g mass

∴ The mass of the pole

= 8 × 111532.8 g

= 892262.4 g

= 892.262 kg

**Question. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of ` 500 per m ^{2}. **

**Solution.**Radius (r) of the cylindrical part = Radius of the conical part = 4/2 m = 2 m Area of the canvas used

= Curved S.A. of the cylindrical part + Curved S.A. of the conical part

**Question. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter hemisphere can have? Find the surface area of the solid. ****Solution.** The figure shows the solid as per the question:

Now the greatest diameter the hemisphere can have

= size of the cube = 7 cm

Surface area of the solid

= Total S.A. of the cube – Area of the circle common to the hemisphere and the cube + Curved

S.A. of the hemisphere

**Question. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see along side). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its: ****(i) surface area (ii) volume****Solution.** Since, diameter of the capsule = 5 mm, its radius = 5/2 mm = 2.5 mm

**Question. A solid is in the form of a cone mounted on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of p. ****Solution.** Let the radius of the hemisphere, r = 1 cm

Then, the height of the cone, h = 1 cm

The volume of the solid

= Volume of the cone + Volume of the hemisphere

**Question. A metallic cylinder has radius 3 cm and height 5 cm. It is made of metal X. To reduce its weight, a conical hole is drilled in the cylinder as shown and it is completely filled with a lighter metal Y. The conical hole has a radius of 3/2 cm and its depth is 8/9 cm. Calculate the ratio of the volume of the metal X to the volume of metal Y in the solid.****Solution.** Volume of metal X in the solid

= Volume of the whole cylinder – Volume of conical hole

**Question. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid. ****Solution.** Given: The diameter of the hemispherical depression = l

**Question. A storage building is in the form as shown in the given figure. The vertical cross-section parallel to the width side of the building is a rectangle of size 7 m × 3 m mounted by a semicircle of radius 3.5 m. The inner measurements of the cuboid are 10 m × 7 m × 3 m. Find the volume of the building and the total internal surface area excluding the floor.****Solution.** Since the storage building is of uniform cross-section

So, its Volume = Its area of cross-section × its length.

**Question. A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter of the spherical part is 8.5 cm. By measuring the amount of water it holds, Rama finds its volume to be 345 cm3. Is she correct by taking the above as the inside measurements? ****Solution.** Radius (r) of the cylindrical neck = 2/2 cm = 1 cm, its height = 8 cm and radius (R) of the spherical part = 8.5/2 cm = 4.25 cm.

Volume of water in the vessel

= Volume of the whole vessel

= Volume of cylindrical neck + Volume of spherical part

**Question. A rectangular metal block has length 15 cm, breadth 10 cm and height 5 cm. From this block, a circular hole of diameter 7 cm is drilled out. Find: ****(i) the volume of the remaining solid (ii) the surface area of the remaining solid. ****Solution.** (i) The volume of the remaining solid

= Volume of rectangular block – Volume of the circular hole

## Mathematics Surface Area And Volume Worksheets for Class 10 as per CBSE NCERT pattern

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