Chapter 12: Surface Areas and Volumes
CBSE Official PYQs (2015-2026)
Q1. If two solid hemispheres of same base radius \(r\) are joined together along their bases, then the total surface area of this new solid is :
Q2. A solid cylinder of radius \(r\) and height \(h\) is placed over other cylinder of same height and radius. The total surface area of the shape so formed is :
Q3. A solid toy is in the form of a hemisphere surmounted by a right circular cone of same base radius. If the radius of the base is \(r\) and height of the cone is \(h\), then the volume of the toy is :
Q4. The volume of the largest right circular cone that can be carved out of a solid cube of side \(a\) is :
Q5. If the radius of the base of a right circular cylinder is halved, keeping its height same, then the ratio of the volume of the reduced cylinder to that of the original cylinder is :
Q6. A metallic spherical shell of internal and external diameters \(4 \text{ cm}\) and \(8 \text{ cm}\) respectively, is melted and recast into a cone of base diameter \(8 \text{ cm}\). The height of the cone is :
Q7. The total surface area of a solid hemisphere of radius \(7 \text{ cm}\) is :
Q8. Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter \(2 \text{ cm}\) and height \(16 \text{ cm}\). The diameter of each sphere is :
Q9. Three solid metallic spheres of radii \(6 \text{ cm}\), \(8 \text{ cm}\) and \(10 \text{ cm}\) respectively are melted to form a single solid sphere. The radius of the resulting sphere is :
Q10. A solid sphere of radius \(r\) is melted and recast into a solid cone of height \(r\). The base radius of the cone is :
Q11. If the curved surface area of a right circular cone of radius \(r\) is \(3\) times the area of its base, then the slant height of the cone is :
Q12. The number of solid spheres of radius \(6 \text{ cm}\) that can be made by melting a solid copper cylinder of radius \(8 \text{ cm}\) and height \(18 \text{ cm}\) is :
Q13. The ratio of the volumes of a cylinder, a cone and a hemisphere of the same base radius and same height is :
Q14. A solid toy consists of a right circular cone placed on a hemisphere of the same base radius \(r\) and slant height \(l\). Find the formula for the total surface area of this solid.
Q15. A rectangular sheet of paper \(40 \text{ cm} \times 22 \text{ cm}\) is rolled to form a hollow cylinder of height \(40 \text{ cm}\). The radius of the cylinder (in cm) is :
Q16. The volume of a solid hemisphere is \(1152 \pi \text{ cm}^3\). Its curved surface area is :
Q17. A cubical block of side \(7 \text{ cm}\) is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have?
Q18. During conversion of a solid from one shape to another, the volume of the new shape will :
Q19. A funnel is the combination of which of the following shapes?
Q20. If the perimeter of one face of a cube is \(20 \text{ cm}\), then its total surface area is :
Q21. If the total surface area of a cube is \(96 \text{ cm}^2\), then the volume of the cube is :
Q22. The surface area of a sphere is \(616 \text{ cm}^2\). Its radius is :
Q23. A solid piece of iron in the form of a cuboid of dimensions \(49 \text{ cm} \times 33 \text{ cm} \times 24 \text{ cm}\) is melted to form a solid sphere. The radius of the sphere is :
Q24. If a cone and a cylinder have equal bases and equal volumes, the ratio of their heights is :
Q25. If the slant height of a cone is \(13 \text{ cm}\) and its base radius is \(5 \text{ cm}\), then the volume of the cone is :
Q26. Find the volume of a right circular cylinder whose base radius is \(10 \text{ cm}\) and height is \(14 \text{ cm}\). (Use \(\pi = \frac{22}{7}\))
Q27. A spherical lead ball of radius \(r\) is melted and recast into 8 identical smaller solid lead balls. The radius of each smaller ball is :
Q28. A cone of height \(24 \text{ cm}\) and radius of base \(6 \text{ cm}\) is made up of modeling clay. A child reshapes it in the form of a sphere. The radius of the sphere is :
Q29. If the volumes of two spheres are in the ratio 64 : 27, then the ratio of their surface areas is :
Q30. The dimensions of a solid metallic cuboid are \(9 \text{ m} \times 8 \text{ m} \times 2 \text{ m}\). It is melted and recast into solid cubes of edge \(2 \text{ m}\). The number of cubes so formed is :
Q31. Assertion (A): If the radius of a cone is doubled and height is halved, its volume remains the same.
Reason (R): Volume of a cone is proportional to the square of radius and height.
Q32. Assertion (A): The total surface area of a solid cylinder is \(2 \pi r (h + r)\).
Reason (R): The curved surface area of a cylinder of radius \(r\) and height \(h\) is \(2 \pi r h\).
Q33. If a solid hemisphere of radius '\(r\)' is melted and recast into a solid right circular cone of base radius '\(r\)', then the height of the cone is :
Q34. A vessel is in the form of a hollow hemisphere. The capacity of the hemisphere of radius \(21 \text{ cm}\) is : (Use \(\pi = \frac{22}{7}\))
Q35. If the slant height and base radius of a cone are \(5 \text{ cm}\) and \(3 \text{ cm}\) respectively, then its curved surface area is :
Q36. The total surface area of a solid hemisphere is \(462 \text{ cm}^2\). Its radius is : (Use \(\pi = \frac{22}{7}\))
Q37. If a solid metallic sphere of radius \(8 \text{ cm}\) is melted and recast into \(n\) identical small spheres of radius \(2 \text{ cm}\) each, then the value of \(n\) is :
Q38. The volume of a solid right circular cone is \(9856 \text{ cm}^3\). If the diameter of its base is \(28 \text{ cm}\), then its height is : (Use \(\pi = \frac{22}{7}\))
Q39. A solid is in the shape of a cone standing 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 \(\pi\).
Q40. Two cubes each of volume \(64 \text{ cm}^3\) are joined end to end. Find the surface area of the resulting cuboid.
Q41. Find the number of plates, \(1.5 \text{ cm}\) in radius and \(0.2 \text{ cm}\) thick, that can be fitted on a solid cylinder of radius \(3 \text{ cm}\) and height \(10 \text{ cm}\).
Q42. A cone of base radius \(4 \text{ cm}\) is melted and recast into a cylinder of same base radius. If height of the cone is \(12 \text{ cm}\), find the height of the cylinder.
Q43. Find the curved surface area of a right circular cone whose slant height is \(25 \text{ cm}\) and base radius is \(7 \text{ cm}\).
Q44. A solid sphere of radius \(10.5 \text{ cm}\) is melted and recast into smaller solid cones, each of radius \(3.5 \text{ cm}\) and height \(3 \text{ cm}\). Find the number of cones so formed.
Q45. A solid metallic cuboid of dimensions \(15 \text{ cm} \times 10 \text{ cm} \times 8 \text{ cm}\) is melted and recast into a cylinder of radius \(7 \text{ cm}\). Find the height of the cylinder (approximate to nearest integer).
Q46. 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. If the radius of each part is \(8 \text{ cm}\), the cylindrical part is \(240 \text{ cm}\) high and the conical part is \(36 \text{ cm}\) high, find the weight of the pillar, given that \(1 \text{ cm}^3\) of iron weighs \(7.5\) grams.
Q47. A vessel is in the form of an inverted cone. Its height is \(8 \text{ cm}\) and the radius of its top, which is open, is \(5 \text{ cm}\). It is filled with water up to the brim. When lead shots, each of which is a sphere of radius \(0.5 \text{ cm}\) are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.
Q48. Find the volume of a sphere whose surface area is \(154 \text{ cm}^2\).
Q49. A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is \(4 \text{ cm}\) and the diameter of the base is \(8 \text{ cm}\). Determine the volume of the toy. (Take \(\pi = 3.14\))
Q50. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are \(2.1 \text{ m}\) and \(4 \text{ m}\) respectively, and the slant height of the top is \(2.8 \text{ m}\), find the area of the canvas used for making the tent.
Q51. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder. If the height of the cylinder is \(10 \text{ cm}\) and its base is of radius \(3.5 \text{ cm}\), find the total surface area of the article.
Q52. A solid cone of radius \(7 \text{ cm}\) and height \(12 \text{ cm}\) is melted and recast into small solid spheres of radius \(0.5 \text{ cm}\). Find the number of spheres so formed.
Q53. A solid toy is in the form of a hemisphere surmounted by a right circular cone of same base radius. If the radius of the base is \(3.5 \text{ cm}\) and total height of the toy is \(15.5 \text{ cm}\), find the total surface area of the toy. (Use \(\pi = \frac{22}{7}\))
Q54. A solid is composed of a cylinder with hemispherical ends. If the whole length of the solid is \(108 \text{ cm}\) and the diameter of the hemispherical ends is \(36 \text{ cm}\), find the cost of polishing its surface at the rate of \(7 \text{ paise}\) per \(\text{cm}^2\).
Q55. A hemi-spherical dome of a temple needs to be painted. If the circumference of the base of the dome is \(17.6 \text{ m}\), find the cost of painting it, given the cost of painting is Rs. 5 per \(100 \text{ cm}^2\).
Q56. A solid metal cone of radius \(12 \text{ cm}\) and height \(24 \text{ cm}\) is melted and recast into solid spheres of radius \(2 \text{ cm}\) each. Find the number of spheres so formed.
Q57. From a solid cylinder of height \(14 \text{ cm}\) and base diameter \(12 \text{ cm}\), a conical cavity of same height and same base diameter is hollowed out. Find the total surface area of the remaining solid. (Use \(\pi = \frac{22}{7}\))
Q58. A hemispherical tank full of water is emptied by a pipe at the rate of \(\frac{25}{7}\text{ litres}\) per second. How much time will it take to empty half the tank, if it is \(3 \text{ m}\) in diameter? (Use \(\pi = \frac{22}{7}\))
Q59. A solid toy is in the form of a hemisphere surmounted by a right circular cone of same base radius. If the radius of the base is \(2.1 \text{ cm}\) and the total height of the toy is \(14.1 \text{ cm}\), find the total volume of the toy. Also, find the cost of painting the toy at the rate of \(\text{Rs. } 10\) per \(\text{cm}^2\), if the slant height of the conical part is \(12.18 \text{ cm}\) (approximate).
Q60. A solid is in the form of a cylinder with hemispherical ends. If the total length of the solid is \(19 \text{ cm}\) and the diameter of the cylinder is \(7 \text{ cm}\), find the total surface area and the volume of the solid. (Use \(\pi = \frac{22}{7}\))
Q61. Case Study 1 (Adventure Camp Organization):
An adventure camp coordinator designed a series of residential tents in the shape of a cylinder surmounted by a conical top of the same base diameter. The diameter of the common base is \(14 \text{ m}\). The height of the cylindrical part is \(11 \text{ m}\) and the total height of the tent is \(35 \text{ m}\).
Based on the above information, answer the following questions:
- Find the slant height of the conical part of the tent. (1 Mark)
- Find the volume of air inside the tent. (2 Marks)
- Find the total surface area of the canvas required to make the tent, if the canvas extends to the ground. (2 Marks)
(i) 25 m
(ii) 2926 m³
(iii) 1034 m²
Q62. Case Study 2 (Rainwater Harvesting System):
A residential housing society created a rainwater harvesting system consisting of a rooftop collection area that channels water into a cylindrical tank with a hemispherical base. The radius of the cylinder is \(2 \text{ m}\) and its height is \(7 \text{ m}\).
Based on the above information, answer the following questions:
- Find the total capacity (volume) of the rainwater harvesting tank. (2 Marks)
- If the tank is filled with water up to a height of \(5 \text{ m}\) in the cylindrical part, find the volume of water currently stored in the tank. (2 Marks)
- Find the total inner surface area of the tank that needs waterproof lining. (1 Mark)
(i) 104.76 m³
(ii) 79.58 m³
(iii) 113.10 m²
Q63. Case Study 3 (Circus Tent Structural Planning):
A circus coordinator planned a giant promotional tent in the shape of a cylinder of diameter \(24 \text{ m}\) surmounted by a conical top of the same diameter. The height of the cylindrical part is \(4 \text{ m}\) and the slant height of the conical top is \(15 \text{ m}\).
Based on the above information, answer the following questions:
- Find the area of the canvas required to make the tent (excluding the floor). (2 Marks)
- Find the total cost of the canvas at the rate of \(\text{Rs. } 200\) per \(\text{m}^2\). (1 Mark)
- Find the volume of space enclosed by the circus tent. (2 Marks)
(i) 867.43 m²
(ii) Rs. 1,73,486
(iii) 2112 m³
Q64. Case Study 4 (Glass Greenhouse Construction):
A botany school constructed a decorative greenhouse. The structure consists of a rectangular cuboidal base of dimensions \(10 \text{ m} \times 8 \text{ m} \times 5 \text{ m}\) surmounted by a half-cylinder of the same width (\(8 \text{ m}\)) and length (\(10 \text{ m}\)).
Based on the above information, answer the following questions:
- Find the volume of air enclosed by the greenhouse structure. (2 Marks)
- Find the total surface area of glass panels required for the walls and the roof of this greenhouse (excluding the floor). (2 Marks)
(i) 525.66 m³
(ii) 345.71 m²