Chapter 12: Surface Areas and Volumes

Important questions — the ones the board asked most (20)

1. Q1Long answer5 marksAsked: 2023, 2024

   A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in the figure. If the height of the cylinder is 5.8 cm and its base is of radius 2.1 cm, find the total surface area of the article.

   

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   Model answer · surface area, combination of solids

   1. 1.Given: height of the cylinder, h = 5.8 cm, and radius of its base, r = 2.1 cm. Each hemisphere scooped out has the same radius, r = 2.1 cm. Take π = 22/7. Scooping out a hemisphere from each end removes both flat ends of the cylinder. What is left on the outside is the curved surface of the cylinder and the inner curved surfaces of the two hemispheres. So, Total Surface Area (TSA) of article = CSA of cylinder + CSA of two hemispheres, where CSA stands for Curved Surface Area. CSA of cylinder = 2πrh = 2 × 22/7 × 2.1 × 5.81 mark
   2. 2.= 2 × 22 × 0.3 × 5.8 = 76.56 cm²1 mark
   3. 3.CSA of one hemisphere = 2πr², so CSA of two hemispheres = 2 × 2πr² = 4πr² = 4 × 22/7 × 2.1 × 2.11 mark
   4. 4.= 4 × 22 × 0.3 × 2.1 = 55.44 cm²1 mark
   5. 5.Total Surface Area of article = CSA of cylinder + CSA of two hemispheres = 76.56 + 55.44 = 132 cm² So, the total surface area of the article is 132 cm².1 mark

   • Write the formula first: "Total Surface Area (TSA) of article = CSA of cylinder + CSA of two hemispheres". The flat ends of the cylinder are not part of it.
   • For the two hemispheres, write "4πr²", which is 2 × 2πr².
   • Write the unit cm² after every area, including the final answer "132 cm²".

   Marks split as in the board’s published marking scheme

2. Q2Long answer5 marksAsked: 2025

   A wooden article was made by scooping out a hemisphere (of same diameter) from one end of a solid cylinder as shown in the given figure. If the height of the cylinder is 10 cm and the diameter of the cylinder is 14 cm, find the total surface area of the remaining wooden article. (Use pi = 22/7)

   

   Medha Ma’am

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   Model answer · surface area of combined solids

   1. 1.Given: height of the cylinder, h = 10 cm, and diameter of the cylinder = 14 cm. So the radius of the cylinder is r = 14/2 = 7 cm. The hemisphere has the same diameter, so radius of hemisphere = radius of cylinder = 7 cm. The hemisphere is scooped out of one end, so that end no longer has a flat base. Instead it shows the inside curved surface of the hemisphere. The other end still has its flat circular base. Total surface area of the remaining article = CSA of cylinder + area of one base of the cylinder + CSA of hemisphere = 2πrh + πr² + 2πr²3 marks
   2. 2.Putting in r = 7 cm, h = 10 cm and π = 22/7: = 2 × 22/7 × 7 × 10 + 22/7 × 7 × 7 + 2 × 22/7 × 7 × 7 = 440 + 154 + 3081.5 marks
   3. 3.= 902 cm² So the total surface area of the remaining wooden article is 902 cm² (902 sq. cm).0.5 marks

   • Write "radius of hemisphere = radius of cylinder = 7 cm" before you use the formula.
   • Write the formula as "2πrh + πr² + 2πr²": count only one base of the cylinder, and use the CSA of the hemisphere in place of the scooped end.
   • End with the unit: "902 cm²".

   Marks split as in the board’s published marking scheme

3. Q3Long answer5 marksAsked: 2025

   A hemispherical depression is scooped out from the top face of a wooden cubical block of side 14 cm. If the diameter of the hemisphere is equal to the side of the cube, find the total surface area of the remaining solid block. (Use pi = 22/7)

   

   Medha Ma’am

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   Model answer · surface area, cube with hemispherical depression

   1. 1.The total surface area of the cubical block is 6 times (edge)^2. Here the edge of the cube is 14 cm, so the total surface area of the cube = 6 × 14 × 14 cm^2 = 1176 cm^2.1 mark
   2. 2.The diameter of the hemisphere is equal to the side of the cube, that is 14 cm. So the radius of the hemisphere r = 14/2 cm = 7 cm.1 mark
   3. 3.The curved surface area of the hemisphere = 2πr^2. Taking r = 7 cm and π = 22/7, the curved surface area of the hemisphere = 2 × 22/7 × 7 × 7 cm^2 = 308 cm^2.1.5 marks
   4. 4.When the hemisphere is scooped out from the top face of the cube, a circular area equal to the base area of the hemisphere, πr^2, is removed from the top surface of the cube. Taking r = 7 cm and π = 22/7, this area = 22/7 × 7 × 7 cm^2 = 154 cm^2.0.5 marks
   5. 5.The total surface area of the remaining solid = total surface area of the cube + curved surface area of the hemisphere – base area of the hemisphere = (1176 + 308 – 154) cm^2 = 1330 cm^2. Therefore, the total surface area of the remaining solid block is 1330 cm^2.1 mark

   Diagram: Draw a cube of side 14 cm with a hemispherical depression scooped out from the centre of its top face, labelling the cube's edge as 14 cm and the hemisphere's radius as 7 cm.

   • Write the formula 6a^2 before substituting to get 1176 cm^2 for the cube's total surface area.
   • State clearly that radius r = 7 cm (half of 14 cm) before using it in further formulas.
   • In the final step, show the addition and subtraction 1176 + 308 – 154 = 1330 and write the unit cm^2 with the answer.

   Marks split from a structure derived from past papers (checked twice); not CBSE’s official scheme

4. Q4Long answer5 marksAsked: 2025

   35. A spherical glass vessel has a cylindrical neck which is 7 cm long and 2 cm in diameter. The diameter of the spherical part is 14 cm. Find the capacity of the entire glass vessel. (Use pi = 22/7)

   How the model answer begins

   1. Given: the glass vessel is a sphere joined to a cylindrical neck. Diameter of the spherical part = 14 cm, so radius of sphere = R = 14/2 = 7 cm. Diameter of the cylindrical neck = 2 cm, so radius of cylinder = r = 2/2 =…

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5. Q5Long answer5 marksAsked: 2025

   A perfume bottle is in the form of a cylinder but the bottom of the bottle has a hemispherical raised portion to reduce the capacity of the bottle. The inner diameter of the bottle is 5 cm and the height of the bottle is 10 cm. Find the capacity of the perfume bottle in mL. (Use pi = 3.14 and 1 cm^3 = 1 mL)

   How the model answer begins

   1. Given: the bottle is a cylinder with a hemispherical raised portion at its bottom. The inner diameter of the bottle is 5 cm, and the hemisphere has the same base as the cylinder. So, radius of cylinder = radius of…

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6. Q6Long answer5 marksAsked: 2025

   A necklace is made up of wooden beads. Each bead is in the form of a sphere of diameter 4.2 mm. A cylinder is hollowed out from each bead. If the radius of the cylinder is 1 mm, find the volume of wood left in each bead.

   How the model answer begins

   1. Given: each bead is a sphere of diameter 4.2 mm, so the radius of a spherical bead = R = 4.2 ÷ 2 = 2.1 mm. The radius of the cylinder hollowed out = r = 1 mm. The cylinder goes right through the bead, so the height of…

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7. Q7Long answer5 marksAsked: 2024

   A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area and the volume of the vessel.

   How the model answer begins

   1. Given: the vessel is a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. The hemisphere and the cylinder have the same radius. Radius r…

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8. Q8Long answer5 marksAsked: 2024

   A tent is in the shape of a cylinder, surmounted by a conical top. If the height and diameter of the cylindrical part are 3.5 m and 6 m, and slant height of the top is 4.2 m, find the area of canvas used for making the tent. Also, find the cost of canvas of the tent at the rate of ₹ 500 per m^2.

   How the model answer begins

   1. Given: the tent is a cylinder with a conical top. Height of the cylindrical part, h = 3.5 m Diameter of the cylindrical part = 6 m Slant height of the conical top, l = 4.2 m The cone and the cylinder share the same…

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9. Q9Long answer5 marksAsked: 2024

   A juice seller was serving his customers using glasses as shown in the figure. The inner diameter of the cylindrical glass was 5.6 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 the apparent capacity and the actual capacity of the glass.

   How the model answer begins

   1. Given: the glass is a cylinder with inner diameter = 5.6 cm and height h = 10 cm. Its bottom has a raised hemisphere with the same radius. Radius (r) = 5.6 ÷ 2 = 2.8 cm

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10. Q10Long answer5 marksAsked: 2024

    A textile industry runs in a shed. This shed is in the shape of a cuboid surmounted by a half cylinder. If the base of the industry is of dimensions 14 m x 20 m and the height of the cuboidal portion is 7 m, find the volume of air that the industry can hold. Further, suppose the machinery in the industry occupies a total space of 400 m^3. Then, how much space is left in the industry ? OR (b) From a solid cylinder of height 8 cm and radius 6 cm, a conical cavity of the same height and same radius is carved out. Find the total surface area of the remaining solid. (Take pi = 3.14)

    How the model answer begins

    1. Given: the shed is a cuboid surmounted by a half cylinder. The base of the shed is 20 m × 14 m and the height of the cuboidal portion is 7 m. The volume of air the shed can hold is the volume of the cuboid plus the…

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11. Q11Long answer5 marksAsked: 2024

    (a) A cloth industry runs in a shed. This shed is in the shape of a cuboid surmounted by a half cylinder. If the base dimensions of the shed are 14 m x 20 m and the height of the cuboidal part is 7 m, find the volume of air that the shed can hold. Further, suppose the machinery in the industry occupies a total space of 400 m^3, then how much space is still left in the shed? OR (b) A conical cavity is carved out of a solid cylinder of height 8 cm and radius 6 cm, such that the height and radius of the cone are the same as the height and radius of the cylinder. Find the total surface area of the remaining solid. [Use pi = 3.14]

    How the model answer begins

    1. Given: the shed is a cuboid surmounted by a half cylinder. The base of the shed is 20 m × 14 m, and the height of the cuboidal portion is 7 m. The volume of air the shed can hold is the volume of the cuboid plus 1/2 the…

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12. Q12Long answer5 marksAsked: 2024

    32. (a) Two cubes each of volume 125 cm^3 are joined end to end. Find the volume and the surface area of the resulting cuboid. OR (b) A solid is in the shape of a cone surmounted on a hemisphere with both their diameters being equal to 7 cm and the height of the cone is equal to its radius. Find the volume of the solid.

    How the model answer begins

    1. Answering part (a). Given: two cubes, each of volume 125 cm^3, are joined end to end to form a cuboid. Volume of one cube = (edge)^3 = 125 cm^3 Edge of the cube = ∛125 = 5 cm So, the side of each cube is 5 cm.

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13. Q13Long answer5 marksAsked: 2024

    (a) Two cubes, each of volume 125 cm^3, are joined end to end. Find the volume and the surface area of the resulting cuboid. or (b) A solid is in the shape of a cone standing on a hemisphere, with both their diameters being equal to 7 cm and the height of the cone is equal to its radius. Find the volume of the solid.

    How the model answer begins

    1. Given: two cubes are joined end to end, and the volume of each cube is 125 cm^3. Let the edge (side) of one cube be a. Volume of one cube = a^3 = 125 cm^3 a = 5 cm So, the side of each cube = 5 cm.

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14. Q14Long answer5 marksAsked: 2024

    (a) A toy is in the form of a cone of radius 7 cm mounted on a hemisphere of the same radius. The total height of the toy is 31 cm. Find the total surface area of the toy. OR (b) A wooden article was made by scooping out a hemisphere from each end of a solid cylinder. If the height of the cylinder is 15 cm and its base radius is 4.2 cm, then find the total surface area of the article so formed.

    How the model answer begins

    1. (a) Given: the toy is a cone of radius 7 cm mounted on a hemisphere of the same radius, and the total height of the toy is 31 cm. Radius of cone = radius of hemisphere = r = 7 cm The height of the hemispherical part…

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15. Q15Long answer5 marksAsked: 2024

    (a) A toy is in the form of a cone of radius 7 cm mounted on a hemisphere of same radius. The total height of the toy is 31 cm. Find the total surface area of the toy. OR (b) A wooden article was made by scooping out a hemisphere from each end of a solid cylinder. If the height of the cylinder is 15 cm and its base radius is 4.2 cm, find the total surface area of the article.

    How the model answer begins

    1. Given: the toy is a cone mounted on a hemisphere, and the base radius of the cone equals the radius of the hemisphere. Radius of cone = radius of hemisphere = r = 7 cm Total height of toy = 31 cm The height of the…

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16. Q16Long answer5 marksAsked: 2023

    From a solid cylinder of height 20 cm and diameter 12 cm, a conical cavity of height 8 cm and radius 6 cm is hallowed out. Find the total surface area of the remaining solid.

    How the model answer begins

    1. The radius of the cylinder is r = 6 cm and its height is h = 20 cm. The radius of the conical cavity is r = 6 cm and its height is h1 = 8 cm.

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17. Q17Long answer5 marksAsked: 2023

    The sum of the radius of the base and height of a solid right-circular cylinder is 37 cm. If the total surface area of the solid cylinder is 1628 cm^2, find the volume of the cylinder.

    How the model answer begins

    1. Given: a solid right circular cylinder whose total surface area is 1628 cm². Let r be the radius of its base and h its height. The sum of the radius and the height is 37 cm, so r + h = 37 cm

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18. Q18Long answer5 marksAsked: 2023

    The boilers are used in thermal power plants to store water and then used to produce steam. One such boiler consists of a cylindrical part in middle and two hemispherical parts at its both ends. Length of the cylindrical part is 7m and radius of cylindrical part is 7/2 m. Find the total surface area and the volume of the boiler. Also, find the ratio of the volume of cylindrical part to the volume of one hemispherical part.

    How the model answer begins

    1. Given: the boiler is a cylinder with two hemispheres stuck at either end. Length of the cylindrical part h = 7 m, and the common radius of the cylinder and each hemisphere r = 7/2 m. Take π = 22/7. Only the curved…

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19. Q19Long answer5 marksAsked: 2025

    A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the top. When some lead shots, each being a sphere of radius 0.5 cm, are dropped into it, one-fourth of the water in the vessel flows out. Find the number of lead shots dropped in the vessel.

    How the model answer begins

    1. The vessel is in the form of an inverted cone with height 8 cm and radius of its open top 5 cm, filled with water up to the top. The volume of a cone is given by the formula (1/3)πr²h. So the volume of water in the…

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20. Q20Long answer5 marksAsked: 2025

    A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. Also, find the surface area of the toy. (Take pi = 3.14)

    How the model answer begins

    1. Given: the toy is a hemisphere surmounted by a right circular cone. Height of the cone (h) = 2 cm Diameter of the base = 4 cm, so Radius of the hemisphere = radius of the base of the cone = r = 1/2 × 4 cm = 2 cm

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57 more questions of this chapter, each with its model answer, in the Ask Medha app.

Objective questions with the board’s key (10)

1. 1. CBSE 2026

   The total surface area of a solid hemisphere of diameter '2d' is :

   • (A) 3 pi d^2 ✓
   • (B) 2 pi d^2
   • (C) 1/2 pi d^2
   • (D) 3/4 pi d^2

   Answer: (A)

2. 2. CBSE 2026

   The volume of a sphere is 36 pi cm^3, then its radius (in cm) is :

   • (A) 3 ✓
   • (B) 3sqrt(3)
   • (C) 3^(2/3)
   • (D) 3^(1/3)

   Answer: (A)

3. 3. CBSE 2026

   A hemispherical bowl is made of steel, whose thickness is 1 cm. The outer radius of the bowl is 6 cm. The volume of the steel used (in cm^3) is :

   • (A) 182 pi
   • (B) 182/3 pi ✓
   • (C) 682/3 pi
   • (D) 364/3 pi

   Answer: (B)

4. 4. CBSE 2026

   Assertion (A) : The surface area of the cuboid formed by joining two cubes of sides 4 cm each, end-to-end, is 160 cm^2. Reason (R): The surface area of a cuboid of dimensions l x b x h is 2(lb + bh + hl).

   • (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). ✓
   • (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
   • (C) Assertion (A) is true, but Reason (R) is false.
   • (D) Assertion (A) is false, but Reason (R) is true.

   Answer: (A)

5. 5. CBSE 2026

   A cone of maximum size is carved out from a solid cube of edge length l. The volume of the cone is :

   • (A) pi * l^3 / 12 ✓
   • (B) pi * l^3 / 3
   • (C) l^3 * (1 - pi/3)
   • (D) pi * l^3 / 8

   Answer: (A)

6. 6. CBSE 2026

   Two wooden solid hemispheres of same radii r, are joined at a point, as shown in the figure. The total surface area of the object is :

   • (A) 4pi r^2
   • (B) 6pi r^2 ✓
   • (C) 3pi r^2
   • (D) 5pi r^2

   Answer: (B)

7. 7. CBSE 2026

   A cone with slant height 10 cm and radius 6 cm is surmounted on a hemisphere of same radius. The height of the toy is :

   • (A) 12 cm
   • (B) 16 cm
   • (C) 14 cm ✓
   • (D) 10 cm

   Answer: (C)

8. 8. CBSE 2026

   A conical cavity of maximum volume is carved out from a wooden solid hemisphere of radius 10 cm. Curved surface area of the cavity carved out is (use π = 3.14)

   • (A) 314 sqrt(2) cm^2 ✓
   • (B) 314 cm^2
   • (C) 3140/3 cm^2
   • (D) 3140 sqrt(2) cm^2

   Answer: (A)

9. 9. CBSE 2026

   A camping tent in hemispherical shape of radius 1.4 m, has a door opening of area 0.50 m^2. Outer surface area of the tent is

   • (A) 11.78 m^2
   • (B) 12.32 m^2
   • (C) 11.82 m^2 ✓
   • (D) 12.86 m^2

   Answer: (C)

10. 10. CBSE 2026

    A cylinder of radius r is surmounted by a hemisphere of same radius. If the total height of this object is 13 cm, then its internal surface area is :

    • (A) 2 pi r(r + 13)
    • (B) 13 pi r
    • (C) 2 pi(13 + r)^2
    • (D) 26 pi r ✓

    Answer: (D)

Questions parents and students ask

How many marks does "Surface Areas and Volumes" carry in CBSE Class 10 Maths?

In the CBSE papers of 2023–2026, this chapter carried about 5 marks and about 2 questions a year; by marks it ranks 7 in Maths.

Which are the most important (VVI) questions of "Surface Areas and Volumes"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 77 written questions of this chapter, with answers, are in the Ask Medha app.

Do the answers follow the board’s marking scheme?

Yes. CBSE publishes its marking schemes; every model answer here is written point by point to the marks the board’s own scheme gives, and checked by a second, independent model.