CBSE Class 10 Maths HOTs Surface Area and Volumes

Surface Area & Volumes

Case Study 1. Ajay is a Class X student. His class teacher Mrs Kiran arranged a historical trip to great Stupa of Sanchi. She explained that Stupa of Sanchi is great example of architecture in India. Its base part is cylindrical in shape. The dome of this stupa is hemispherical in shape, known as Anda. It also contains a cubical shape part called Hermika at the top. Path around Anda is known as Pradakshina Path. Based on the above information, answer the following questions.

(i) Find the lateral surface area of the Hermika, if the side of cubical part is 8 m.
Answer : 256 m²

(ii) The diameter and height of the cylindrical base part are respectively 42 m and 12 m. If the volume of each brick used is 0.01 m³, then find the number of bricks used to make the cylindrical base.
Answer : 16,63,200 bricks

(iii) Find the Curverd surface area of Anda if its radius is 21m.
Answer : 882 𝜋 cm²

CASE STUDY 2. To make the learning process more interesting, creative and innovative, Amayras class teacher brings clay in the classroom, to teach the topic - Surface Areas and Volumes. With clay, she forms a cylinder of radius 6cm and height 8 cm. Then she moulds the cylinder into a sphere and asks some questions to students.

(i) The radius of the sphere so formed is ---------------
Answer : 6cm

(ii) The volume of the sphere so formed is ---------------
Answer : 288 𝜋 cm³

(iii) Find the ratio of the volume of sphere to the volume of cylinder.
Answer : 1:1

(iv) Total surface area of the cylinder is ------------------
Answer : 28 𝜋 cm²

(v) During the conversion of a solid from one shape to another the volume of new shape will --------------------
Answer : Same

1 marks Questions

Question. What is the surface area of a cube whose volume is 64 cm³ ?
Answer: 16Π cm²

Question. A wooden solid sphere of radius r cm is divided into two equal parts. What is the whole surface area of the two parts?
Answer: 3Π r² cm²

Question. If the curved surface area of a right circular cylinder is 1760 cm2 and its radius is 21 cm, then what is its height?
Answer: 13.3 cm

Question. Two cubes each of volume 64 cm3 are joined face to face. What is the surface area of theresulting cuboid ?
Answer: 160 cm2

Question. How many balls, each of radius 1 cm, can be made from a solid sphere of lead of radius 8 cm ?
Answer: 512 

Question. The diameter of a metallic sphere is 6 cm. It is melted and drawn into a wire having diameter of  the cross-section as 0.2 cm. Find the length of the wire?
Answer: 9 m

Question. The base radii of 2 right circular cones of the same height are in the ratio 3:5. Find the ratio of their volumes.
Answer: 9:25

Question. The circumference of the base of a 9m high wooden solid cone is 44m. Find its volume.
Answer: 462cm³

Question. A petrol tank is a cylinder of base diameter 21 cm and length 18 cm fitted with a conical end of length 9 cm. Determine the capacity of the tank.
Answer: 29106 cm³

Question. A rocket is in the form of a cylinder, closed at the lower end, has a cone attached to its top. If each one has a radius 20 cm and height 21 cm, find the surface area of the rocket.
Answer: 5720 cm²

Question. The radius and height of a solid right circular cone are in the ratio of 5 : 12. If its volume is 314 cm3, find its total surface area. (Use 𝜋=3.14 )
Answer: 266.9 cm²

Question. A cone of maximum size is carved out from a cube of edge 14 cm. Find the surface area of the solid left out after the cone is carved out.
Answer: 1365.2 cm²

Question. A hollow cylindrical pipe is made up of copper. It is 21 dm long. The outer and inner diameters of the pipe are 10cm and 6cm respectively. Find the volume of copper used in making the pipe (π= 22/7)
Answer: 10560cub.cm

Question. If the radius of the base of a right circular cylinder is halved, keeping the height same, find the ratio of the volume of the reduced cylinder to that of the original cylinder.
Answer: 1: 4

Question. A tent is in the shape of a cylinder of diameter 20m and height 2.5cm, surmounted by a cone of equal base and height 7.5m. find the capacity of the tent.(take π = 3.14)
Answer: 1570m²

Question. A vessel in the shape of a hollow hemi-sphere mounted by a hollow cylinder. The diameter of the hemi-sphere is 14cm and the total height of the vessel is 13cm. find the inner surface area of the vessel.
Answer: 572cm²

Question. A solid is in the shape of a cone surmounted on a hemisphere. The radius of each of them is 3.5 cm and the total height of the solid is 9.5 cm. Find the volume of the solid.
Answer: 166.83 cm³

Question. A solid cylinder of radius r and height h is placed over another cylinder of same height and radius. Find the total surface area of the shape so formed
Answer: 𝟐𝝅𝒓^𝟐 + 4πrh sq. units

Question. If h, c and V respectively are the height, curved surface area and volume of a cone then find 3𝜋Vh³ – c²h² + 9V² = ……………….
Answer: 0

Question. A conical vessel whose inner radius is 10cm and height 48cm is full of water. Find the volume of water in it.
Answer: 5024cm²

Question. Fifty circular plates each of radius 7cm and thickness 0.5cm are placed one above another to form a solid right circular cylinder. Find its TSA.
Answer: 1408cm²

Question. What is the height of a cone whose base area and volume are numerically equal?
Answer: 3 units 

Question. A rectangular piece of paper of dimensions 60 cm X 88 cm is rolled to form a hollow circular cylinder of height 60 cm. What is the radius of the cylinder?
Answer: 14 cm

Question. In the adjoining figure, the bottom of the glass has a hemispherical raised portion. The glass is fitted with orange juice. How much juice a person will get? (9)
Answer: 396Π cm³

Question. Two right circular cones X & Y are made, X having three times the radius of Y and Y having half the volume of X. Calculate the rates of X & Y?
Answer: H/h=1/18

2/3 marks Questions

Question. The curved surface area of a cone is 550 cm2. Find the volume of the cone given that its base diameter is 14 cm.
Answer: 1334.66 cm³

Question. A square field and an equilateral triangular park have equal perimeters. If the cost of ploughing the field at the rate of Rs. 5 per m2 is Rs. 720, find the cost of maintaining the park at the rate of Rs. 10 per m².
Answer: Rs. 4428.80

Question. An iron solid sphere of radius 3 cm is melted and recast into small spherical balls of radius 1 cm each. Assuming that there is no wastage in the process, find the number of small spherical balls made from the given sphere?
Answer: 27

Question. A rectangular water tank of base 11 cm X 6 cm contain water up to height of 5 m. If the water in the tank is transferred to a cylindrical tank of radius 3.5 m, find the height of water level to be raised in this tank?
Answer: 8.57m

Question. The base radii of two circular cones of the same height are in the ratio 3:5. Find the ratio of their volumes.
Answer: 9:25

Question. Circumference of the edge of hemispherical bowl is 132 cm. Find the capacity of the bowl.( π = 22/7)
Answer: 19104 cm³

Question. In the given figure, a cone of radius 10 cm is divided into two parts by drawing a plane through the mid-points of its axis, parallel to its base. Compare the volume of the two parts? 

Answer: 1:7

Question. Find the volume of the largest right circular cone that can be cut out of a cube whose radius is 9 cm.
Answer: 190.93 cm³

Question. 50 circular plates, each of radius 10.5 cm and thickness 1.6 cm, are placed one above the other to form a solid circular cylinder. Find the curved surface are and volume of the cylinder so formed?
Answer: (5280 cm², 27720 cm³)

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 diameter of the base of the cylinder.
Answer: r=3.5 cm, d=7 cm

STATISTICS(HOTS)

Q1. The given distribution shows the number of runs scored by some top batsman of the world in one day international cricket matches. 

Find the mode of the data.

Q2. Give the empirical relation between median, mode and mean.

Q3. If the median of the distribution given below is 28.5. Find the value of ‘x’ and ‘y’. 

Q4. Find the median of the following frequency distribution 

Q5. The median of the following data is 20.75. Find the missing frequencies ‘x’ and ‘y’ . If the total frequency is 100.  

Statistics (Hots)

(1) 4608.7 runs
(2) Mode = 3 median – 2 Mean
(3) (x = 8/y = 8)
(4) 525
(5) y = 20 , x = 17
(6) P = 7 

Q. 48  Find the area of the largest triangle inscribed in a semi-circle of radius 8 cm.

Q. 49  A semi circular region & a square region have equal perimeters. The area of the square region exceeds that of the semicircular region by 4m2 . Find the perimeter & areas of the two regions.

6 marks questions ( are under HOTS)

Q. 50 Find the ratio of outside & inside curved surface areas of a hollow cylindrical metallic pipe 14 cm long if the difference between the two radii is 1/2 cm. The pipe is made of 99cm3 metal.

Q. 51 A student Kartikai, developed a model of a hut. He made a cylinder of radius 6 cm, & height 6 cm and attached a hemisphere of radius 7 cm at its top. He left an open area for door on the

cylinder of length 4 cm & breadth 3 cm. Find the surface area of the hut.

Q. 52 Dinesh, a civil engineer constructed shape on the entrance of a building as shown in the figure. The radius of sphere is 15 m. The height & radius of cone are 90 cm & 30 cm respectively.

Find the slant height of cone and cost of construction at the rate Rs 3 per cm3 . (Use √10 = 3.16 )

Q. 53 A cylindrical piece of wax of radius 2.8 cm & height 2 cm is placed in a metallic hemispherical bowl of diameter 7 cm. If the wax melts into the bowl, will the bowl overflow. If yes, how much wax overflows. If not, how much more can be accomodated..

Q. 54 A vessel is in the form of an inverted cone. Its height is 8 cm & radius of the top, which is open, is 5 cm. It is filled with water up to the brim. When solid balls of metal of radius 1/2cm are dropped into the vessel, one fourth of the water flows out. Find the number of balls dropped into the vessel.

Q. 55 Water flows at the rate of 10m/minute from a circular pipe of diameter 1/2cm . How much time will it take to fill a vessel which is an inverted cone of radius 20 cm & height 24 cm.

Q. 56 Water is flowing, through a circular pipe of internal diameter 2 cm into a cylindrical tank of base radius 40 cm, at the rate of 7m/second. Find the rise in water level in half on hour.

Q. 57 A solid spherical ball of the of metal is divided into two hemispheres and joined as shown in the figure. This solid is placed in a cylindrical tub, full of water in such a way that the whole solid is sub merged in water. The radius & height of cylindrical tub are 4 cm & 5 cm respectively. The radius of spherical ball is 3 cm. Find the volume of water left in the cylindrical tub.

Q. 58 A metallic toy is made in the form of a hemisphere surmounted by a right cone whose circular base coincides with the plane surface of the hemisphere. The radius of the cone is 3.5 m & its volumes is 2/3 rd of the hemisphere. Calculate the height of cone & the surface area of the toy.

Q. 59 The height of a solid cylinder is 15 cm & diameter 7 cm. Two equal conical holes of radius 3 cm & height 4 cm are cut off as shown in figure. Find the volume and surface area of remaining solid.

Q. 60 The capacity of an ice-cream container is 4 litre. This is to be served to the guests in cups of inverted frustum shape of upper & lower radii 3.5 cm, 2.1 cm and height 4.2 cm. How many containers (approximately) are required for a gathering of 150 people.

Q. 61 A solid spherical ball of copper of radius 5 cm has been melted to form two paper weights of conical shape of equal heights 4 cm, but the radius of one is double of the other. Find the radii of the two paper weights.

Q. 62 The circular ends of a bucket are 20 cm & 14 cm. Its height is 48 cm. Find the ratio of total surface area of the two cones generated by taking both the circular ends as base one by one.

Q. 63 The slant height of right circular cone is 10 cm & its height is 8 cm. It is cut by a plane parallel to its base passing through the mid point of the height. Find ratio of the volumes of two parts.

Q. 64 A spherical ball of copper is melted & made into smaller balls of half the radius of the original. How many such balls can be made? Also find the ratio of total surface area of all the similar balls to that of the original one.

Q. 65 A plane is drawn parallel to the base of a right circular cone so that it divides the cone into two parts of equal surfaces. In what ratio the plane divideds the height of the cone?

Q. 66 A cone of height H is cut by a parallel plane at a distance H/3 from the base. Find the percentage of volume of the frustum produced, of the original cone.

Q. 67 The upper portion of a right circular cone of height h is cut off by a plane parallel to the base & is removed. If the curved surface area of remainder be 3/ 4 of curved surface of the whole one. Find how height is the cutting plane from the base of cone.

Q. 68  An oil tanker, cylindrical in shape, having diameter 2 m & length 4.2 m supplies oil to the two places in the ratio 3 : 2. One of the two places has a rectangular vessel having base area 3.96m² & the other has a cylindrical vessel having diameter 2m. Find the level of oil in each of the two vessels.

Q. 69  Ashok wants to cover his circular wheat granary with a fibre sheet to make it air tight so that no open space is left uncovered as shown in the figure. Cone is of height 4 m & base radius 3 m. The radius & height of cylindrical wall of fibre sheet are 2m & 1m. Find the total area of fibre sheet required to built such cover.

Q. 70 Tosh & Akul went to an ice-cream parlour. Tosh orders double scoop ice-cream & Akul for softy cone as shown in figure . The height of softy cone is 12cm & internal radius is 5/2cm The radius of scoops 2½ cm. Who got more quantity of ice-cream & by how much if ice-cream in the softy cone is up to the brim.

Q. 71  A bucket is of capacity 48510 cm3 . The radii of two circular ends are 28 cm & 7 cm. Find the height of bucket & cost of metal required to make the bucket at the rate of Rs 1 per sq. cm. Find the cost of milk at the rate of Rs 20 per litre. (In this question if we replace radii by perimeters 176 cm & 44cm then solve this problem. Answer will be the same).

Q. 72  A conical vessel of radius 6 cm & height 8 cm is completely filled with water. A solid sphere is immersed into it & its size is such that when it touches the sides, it is just immersed. Find the quantity of water overflows?

Q. 73  Two maths teachers observed six pillars to support shades of metro-station at Kashmiri-Gate which are of the shape as given in the figure. One of them said that the approximately.. height & radius of cylindrical portion are 1m & 1/2 m respectively. The total height of the pillar is approximately.3 * 1/2 m & the radius of the upper part is 1m. She asked the other teacher how much money is required to construct one such pillar if the current rate of construction is Rs. 3000 per cubic metre.

Q. 74  An orange contains juice about 15% of its volume. Find approximately how many dozen oranges are required for a gathering of 50 people if each guest is to be served with 250 ml juice. Assuming that radius of each orange is 3.5 cm.

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

2. Two cubes each of volume 64 cm3 are joined face to face. What is the surface area of the resulting cuboid?

3. A piece of paper is in the form of a semicircular region of radius 10 cm .It is rolled to form a right circular cone. What is the radius of the cone?

4. The largest possible sphere is carved out of a cube of wood of side 21 cm; find the volume of the remaining wood.

5. A sphere and a cube have the same surface area. Show that the ratio of the volume of the sphere to that of cube is √6: √p .

6. If h, Cand V respectively are the height, the curved surface area and volume of the cone. Prove that 3πVh3-C²h²+9V²=0.

7. Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a rectangular tank which is 50 m long and 44m wide. Find the time in which the level of water in the tank will rise by 21cm?

8. A hollow cone is cut by a plane parallel to the base and upper portion is removed. If the curved surface area of the remainder is 8/9 of the curved surface area of the whole cone, Find the ratio of the line segment into which the altitude of the cone is divided by the plane.

9. Two spheres of the same metal weight 1 kg and 7 kg. The radius of the smaller sphere is 3cm. The two spheres are melted to form a single big sphere. Find the diameter of the big sphere.

10. A Copper wire 3mm in diameter is wound about a cylinder of length 120 cm and diameter 10 cm, so as to cover the curved surface of cylinder. Find the length and mass of the wire assuming the density of copper to be 8.88 gm/cm³.

11. A circus tent is made of canvas, and is in the form of a right circular cylinder and a right circular cone above it. The Diameter and height of the cylindrical part of the tent are 126 meter and 5 meter respectively. The total height of the tent is 21 meter. Find the total cost of the canvas used to make the tent when the cost per square meter of the canvas is Rs 12. {Take π=22/7}

12. A bucket is in the form of frustum of a cone with a capacity of 12308.8 cm³ of water.The radii of a top and bottom circular ends are 20 cm and 12 cm respectively. Find the height of the bucket and area of the metal sheet used in its making. [Take π =3.14]

13. A toy is in the shape of right circular cylinder with the hemisphere on one end and a cone on the other end. The height and 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 the height of the conical part is 30 cm. [Take π =3.14]

14. The surface area of a sphere and a cube are equal. Prove that their volumes are in the ratio 1:√(∏/6)

15. A sector of a circle of radius 15cm has the angle 1200. It is rolled up so that two bounding radii are joined together to form a cone. Find the volume of the cone.

16. A cylindrical container of radius 6cm and height 15cm is filled with ice-cream. The whole ice-cream is distributed among 10 children in equal cones having hemispherical top. If the height of the conical portion is four times the radius of the base find the radius of the base.

ANSWER

3. Circumference of the semicircle= circumference of the base of the cone.

6. Assume the radius as r and slant height as l; Substitute the values of C and V in the equation in terms of r, h and l.

7. Volume of water flowing through the pipe per unit time = area of its cross section ×speed of water.

8. ΔAOD~ΔAPB. surface area of thel e cone / surface area of the small cone = 1/9

9. Mass=Volume×density; Ratio of volumes of two spheres=ratio of their mass.

10. Number of turns=height of the cylinder÷diameter of the wire. Length of the wire=No. of turns ×Circumference of the base of the cylinder.

11. Area of the canvas = CSA of the cone + CSA of the cylinder.

12.Apply the formula of volume of the frustum of a cone to get the height of the bucket. Area of the metal sheet= CSA of the bucket + area of the base.

13. Total surface of the toy=CSA of the cylinder + CSA of hemisphere + CSA of the cone.

15. Area of the sector= CSA of the cone. Radius of the sector= slant height of the cone.

16. Volume of the ice cream cone with hemispherical top = volume of the cone + volume of hemisphere.