Surface Areas and Volumes – important questions for Boards 2027
CBSE Class 10 Maths
An object made by joining a cylinder and two hemispheres is an example of: (A) a single basic solid (B) a combination of basic solids (C) a plane figure (D) a frequency distribution.
Answer
(B) a combination of basic solids — it is made by joining a cylinder and two hemispheres.
When finding the surface area of a solid formed by joining two solids, what should be done with the surfaces that become hidden at the join?
Answer
Exclude the surfaces hidden where the solids join; include only exposed surfaces in the total surface area. For a cylinder with hemispheres at both ends, count the cylinder’s curved surface and both hemispheres’ curved surfaces.
A solid consists of a cylinder of radius 3 cm and height 8 cm surmounted by a hemisphere of radius 3 cm. Find its volume. Use π = 3.14.
Answer
Given: Cylinder radius r = 3 cm; cylinder height h = 8 cm; hemisphere radius = 3 cm; π = 3.14. Formula: Total volume = volume of cylinder + volume of hemisphere. Substitution: Total volume = πr²h + ⅔πr³. Cylinder volume = 3.14 × 3² × 8 = 226.08 cm³. Hemisphere volume = ⅔ × 3.14 × 3³ = 56.52 cm³. Total volume = 226.08 + 56.52 = 282.6 cm³. Answer: The solid’s volume is 282.6 cm³.
A solid consists of a cylinder of radius 3 cm and length 10 cm, with a hemisphere attached to each circular end. Find the total surface area of the solid. Use π = 3.14.
Answer
Given: The radius of the cylinder and each hemisphere is r = 3 cm. The length of the cylinder is h = 10 cm. Formula: Total surface area = curved surface area of cylinder + curved surface areas of both hemispheres. Only the curved surfaces are exposed because the circular faces are joined together. The two hemispheres together have the curved surface area of one sphere, 4πr². Therefore, total surface area = 2πrh + 4πr². Substitution: Total surface area = 2 × 3.14 × 3 × 10 + 4 × 3.14 × 3². The curved surface area of the cylinder is 2 × 3.14 × 3 × 10 = 188.4 cm². The curved surface area of both hemispheres is 4 × 3.14 × 3² = 113.04 cm². Adding the exposed areas gives 188.4 + 113.04 = 301.44 cm². Answer: The total surface area of the solid is 301.44 cm².
A cylindrical glass has an inner radius of 3 cm and height 10 cm. A hemispherical raised portion of radius 3 cm occupies part of its base. Find the glass’s actual capacity. Use π = 3.14.
Answer
Given: Inner radius of the glass, r = 3 cm; height, h = 10 cm; π = 3.14. Formula: Apparent capacity = volume of cylinder = πr²h. Substitution: Apparent capacity = 3.14 × 3² × 10 cm³. Calculation: Apparent capacity = 3.14 × 9 × 10 cm³ = 282.6 cm³. The raised hemispherical portion occupies space inside the glass and reduces its actual capacity. Formula: Volume of hemisphere = ⅔πr³. Substitution: Volume of hemisphere = ⅔ × 3.14 × 3³ cm³. Calculation: Volume of hemisphere = ⅔ × 3.14 × 27 cm³ = 56.52 cm³. Formula: Actual capacity = apparent capacity − volume of hemisphere. Substitution: Actual capacity = 282.6 cm³ − 56.52 cm³. Answer: The actual capacity of the glass is 226.08 cm³.
A long container with rounded ends can be modelled as: (A) a cone with two hemispheres (B) a cylinder with a hemisphere at each end (C) a cube with two cones (D) a sphere joined to a cuboid.
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A cone and a hemisphere are joined along equal-radius flat faces. Which surfaces contribute to the total surface area of the resulting solid?
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A toy consists of a cone surmounted by a hemisphere of radius 5 cm. Its total height is 17 cm. Find the cone’s height, its slant height, and the total surface area of the toy. Use π = 3.14.
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A solid toy is made from a cylinder of radius 3 cm and height 10 cm with a cone of the same radius and height 4 cm attached at one end. Find the volume of the toy. Use π = 3.14.
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How is the volume of air in an enclosed space found when objects and people occupy part of it?
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5 more probable questions in this chapter, every one with a full answer.
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