Chapter 13: Surface Areas and Volumes

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

1. Q1Long answer5 marksAsked: 2025

   The slant height of a frustum of a cone is 4 cm and the perimeters (circumferences) of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum.

   

   Medha Ma’am

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   Model answer · curved surface area, frustum of cone

   1. 1.The slant height of the frustum of the cone is given as l = 4 cm. Let r1 and r2 be the radii of the two circular ends of the frustum. We need to find the curved surface area of the frustum, which requires the values of r1, r2 and l.1 mark
   2. 2.The circumferences of the circular ends of the frustum are given as 2πr1 = 18 cm and 2πr2 = 6 cm, where r1 and r2 are the radii of the two circular ends.1 mark
   3. 3.Adding the two circumferences, we get 2πr1 + 2πr2 = 18 + 6 = 24 cm. So, 2π(r1 + r2) = 24. Therefore, the sum of the radii is r1 + r2 = 24/2π = 12/π cm.1 mark
   4. 4.The curved surface area of the frustum of the cone is given by the formula π(r1 + r2)l, where r1 and r2 are the radii of the two circular ends of the frustum and l is the slant height of the frustum.1 mark
   5. 5.Given: r1 + r2 = 12/π cm, l = 4 cm Formula: Curved surface area = π(r1 + r2)l Substitution: Curved surface area = π × (12/π) × 4 Calculation: Curved surface area = 12 × 4 The curved surface area of the frustum = 48 cm21 mark

   • Write the formula 'curved surface area of the frustum of the cone = π(r1 + r2)l' exactly as given before substituting values.
   • Show the step 2πr1 + 2πr2 = 24 leading to r1 + r2 = 12/π cm as a separate working line.
   • End the numerical with the final answer stated with its unit: 48 cm2.

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

2. Q2Long answer5 marksAsked: 2022

   A cone is of height 24 cm and radius of its base is 6 cm. It is melted to form a sphere. Find the radius of the sphere.

   

   Medha Ma’am

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   Model answer · volume, cone melted into sphere

   1. 1.When a solid cone is melted and reshaped into a sphere, the volume of the clay remains the same before and after reshaping. The volume of a cone is given by the formula 1/3 × π × r² × h, where r is the radius of the base and h is the height of the cone.1 mark
   2. 2.Given height of the cone h = 24 cm Given radius of the cone r = 6 cm Formula: Volume of cone = 1/3 × π × r² × h Substitution: Volume of cone = 1/3 × π × 6² × 24 Calculation: Volume of cone = 1/3 × π × 36 × 24 Volume of cone = 288π cubic cm1 mark
   3. 3.If R is the radius of the sphere formed, then its volume is given by the formula 4/3 × π × R³.1 mark
   4. 4.Since the volume of clay in the form of the cone and the sphere remains the same, the volume of the cone is equated to the volume of the sphere. Equation: 4/3 × π × R³ = 288π1 mark
   5. 5.Formula: 4/3 × π × R³ = 288π Simplifying: R³ = 288 × 3/4 Calculation: R³ = 216 R³ = 6³ Therefore, the radius of the sphere R = 6 cm1 mark

   • Write the formula 1/3 × π × r² × h before substituting values for the cone's volume.
   • Show the equation 4/3 × π × R³ = 288π clearly before solving for R.
   • State the final answer as 'radius of the sphere is 6 cm' with the unit.

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

3. Q3Long answer5 marksAsked: 2022

   An article is made by scooping out a hemisphere from each end of a solid wooden cylinder. If the height of the cylinder is 10 cm and its base radius is 3.5 cm then find the surface area and volume of the remaining solid.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 200 s

   Model answer · surface area and volume, combination of solids

   1. 1.Surface area of the remaining solid = CSA of the cylinder + CSA of the two hemispheres = 2πrh + 2 × 2πr² (r = 3.5 cm, h = 10 cm).1 mark
   2. 2.CSA of the cylinder = 2 × 22/7 × 3.5 × 10 = 220 cm²; CSA of the two hemispheres = 4 × 22/7 × 3.5 × 3.5 = 154 cm².1 mark
   3. 3.So the surface area = 220 + 154 = 374 cm².1 mark
   4. 4.Volume = πr²h − 2 × (2/3)πr³: cylinder = 22/7 × 3.5 × 3.5 × 10 = 385 cm³; two hemispheres = 4/3 × 22/7 × (3.5)³ ≈ 179.67 cm³.1 mark
   5. 5.So the volume of the remaining solid = 385 − 179.67 ≈ 205.33 cm³ (616/3 cm³).1 mark

   Diagram: Draw a horizontal cylinder of height 10 cm and base radius 3.5 cm with a hemisphere of radius 3.5 cm scooped out from each flat circular end, labelling h = 10 cm and r = 3.5 cm on the figure.

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

4. Q4Long answer5 marksAsked: 2022

   A solid cube of edge 22 cm is melted to form a circular cylinder of length 7 cm. Find the radius of cylinder.

   How the model answer begins

   1. The volume of the cube is the cube of the side. Side of cube = 22 cm Volume of cube = 22 × 22 × 22 cubic cm

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5. Q5Long answer5 marksAsked: 2023

   A well of diameter 3 m is dug 14 m deep. The soil 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.

   How the model answer begins

   1. The radius of the well is r = 1.5 m, since the diameter of the well is 3 m. The depth of the well is h = 14 m, as the well is dug 14 m deep.

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6. Q6Long answer5 marksAsked: 2023

   A well of diameter 3 m is dug 14 m deep. The soil 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.

   How the model answer begins

   1. The diameter of the well is 3 m, so the radius of the well is r = 3/2 = 1.5 metres. The depth of the well is h = 14 metres.

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7. Q7Long answer5 marksAsked: 2021

   A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Find the volume of the toy.

   How the model answer begins

   1. The radius of the cone is r = 3.5 cm, which is the same as the radius of the hemisphere, since the cone is mounted on the hemisphere of the same radius. The total height of the toy is 15.5 cm, and this total height is…

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8. Q8Short answer2 marksAsked: 2025

   The diagonal of a cube is 9sqrt(3) cm. Find the total surface area of cube.

   How the model answer begins

   1. The diagonal of a cube of side a is given by the formula diagonal = √3 × a (a√3).

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9. Q9Short answer2 marksAsked: 2025

   If the radius of base of a cone is 7 cm and its height is 24 cm then find its curved surface area.

   How the model answer begins

   1. Given the radius of the base of the cone r = 7 cm and the height h = 24 cm. The slant height of the cone is l = √(r² + h²) = √(7² + 24²) = √(49 + 576) = √625 = 25 cm.

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10. Q10Short answer2 marksAsked: 2026

    A metallic cuboid whose length, breadth and height are 100 cm, 80 cm and 64 cm respectively, is melted and recast into a cube. Then find the length of the edge of the cube.

    How the model answer begins

    1. When a metallic cuboid is melted and recast into a cube, the volume remains the same. The volume of the cuboid = length × breadth × height = 100 × 80 × 64 cubic cm = 512000 cubic cm.

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11. Q11Short answer2 marksAsked: 2026

    The slant height of a frustum of a cone is 4 cm and the perimeters of its circular ends are 15 cm and 6 cm respectively. Find the curved surface area of the frustum.

    How the model answer begins

    1. The curved surface area of the frustum of the cone is found using the formula π(r1 + r2)l, where r1 and r2 are the radii of its two circular ends and l is the slant height of the frustum.

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12. Q12Short answer2 marksAsked: 2026

    The volume of a right circular cone is 100pi cm^3 and height is 12 cm, then find the radius of right circular cone.

    How the model answer begins

    1. The volume of a cone is given by the formula V = 1/3 × π × r² × h, where r is the radius of the base and h is the height of the cone.

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13. Q13Short answer2 marksAsked: 2022

    The volume of a right circular cylinder of length 14 cm is equal to the volume of a cube made on an edge of 11 cm. Find the radius of the cylinder.

    How the model answer begins

    1. The volume of a right circular cylinder of radius r and height h is πr²h, and the volume of a cube of edge a is a³. Since the volume of the cylinder of height 14 cm and radius r is equal to the volume of the cube of…

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14. Q14Short answer2 marksAsked: 2022

    How many solid spheres of radius 1 cm can be made from a solid sphere of radius 8 cm ?

    How the model answer begins

    1. The volume of the solid sphere of radius 8 cm is given by the formula for the volume of a sphere, 4/3 * π * r³. Here the radius R = 8 cm, so the volume of the large sphere = 4/3 * π * (8)³ = 4/3 * π * 512 cm³.

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15. Q15Short answer2 marksAsked: 2022

    Find the curved surface area of a hemisphere of radius 3.5 cm.

    How the model answer begins

    1. The formula for the curved surface area of a hemisphere of radius r is 2πr².

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16. Q16Short answer2 marksAsked: 2022

    Two cubes of 8 cm side are joined equally. Find the total surface area of resulting cuboid.

    How the model answer begins

    1. When two cubes of side 8 cm are joined end to end, the resulting solid is a cuboid. Its length becomes 8 + 8 = 16 cm, while its width remains 8 cm and its height remains 8 cm, since these two faces are unchanged by the…

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17. Q17Short answer2 marksAsked: 2023

    Find the volume of a sphere whose radius is 7 cm.

    How the model answer begins

    1. The formula for the volume of a sphere is V = 4/3 πr³, where r is the radius of the sphere.

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18. Q18Short answer2 marksAsked: 2023

    Find the volume of a sphere whose radius is 7 cm.

    How the model answer begins

    1. The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius of the sphere.

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19. Q19Short answer2 marksAsked: 2021

    If the curved surface area of a cylinder is 1760 cm^2 and the radius of its base is 14 cm, then what will be the height of the cylinder?

    How the model answer begins

    1. The formula for the curved surface area of a cylinder is 2πrh, where r is the radius of the base and h is the height of the cylinder.

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20. Q20Short answer2 marksAsked: 2021

    If the radius of the base of a cone is 7 cm and its height is 14 cm, then what will be its curved surface area?

    How the model answer begins

    1. The curved surface area of a cone of base radius r and slant height l is given by the formula πrl.

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2 more questions of this chapter, each with its model answer, in the AskMedha app.

Objective questions with the board’s key (10)

1. 1. Bihar Board 2026

   Three cubes having edge 6 cm are joined end to end to form a cuboid. Then its resultant volume is

   • (A) 648 cm^3 ✓
   • (B) 216 cm^3
   • (C) 432 cm^3
   • (D) 864 cm^3

   Answer: (A)

2. 2. Bihar Board 2026

   If the total surface area of a cube is 864 sq.cm then its volume is

   • (A) 1728 cubic cm ✓
   • (B) 3456 cubic cm
   • (C) 864 cubic cm
   • (D) 432 cubic cm

   Answer: (A)

3. 3. Bihar Board 2026

   If the surface areas of two spheres are in the ratio of 16 : 9 then the ratio of their volumes is

   • (A) 27 : 64
   • (B) 64 : 27 ✓
   • (C) 4 : 3
   • (D) 3 : 4

   Answer: (B)

4. 4. Bihar Board 2026

   If the areas of three adjacent faces of a cuboid are x, y and z respectively then the volume of the cuboid is

   • (A) xyz
   • (B) sqrt(xyz) ✓
   • (C) x^2y^2z^2
   • (D) x^3y^3z^3

   Answer: (B)

5. 5. Bihar Board 2026

   The height of a cylinder is 14 cm and its curved surface area is 264 sq.cm ; then its volume is

   • (A) 1848 cm^3
   • (B) 1232 cm^3
   • (C) 396 cm^3 ✓
   • (D) 308 cm^3

   Answer: (C)

6. 6. Bihar Board 2026

   Area of four walls of a room whose length, breadth and height are l, b and h units respectively, is

   • (A) (l + b) h sq.unit
   • (B) lbh sq.unit
   • (C) 2(l + b)h sq.unit ✓
   • (D) 4(l + b)h sq.unit

   Answer: (C)

7. 7. Bihar Board 2026

   The volume of a sphere of radius R is

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

   Answer: (C)

8. 8. Bihar Board 2026

   If the length of the diagonal of a cube is 5 sqrt(3) cm then its surface area is

   • (A) 120 cm^2
   • (B) 140 cm^2
   • (C) 150 cm^2 ✓
   • (D) 250 cm^2

   Answer: (C)

9. 9. Bihar Board 2026

   What is the ratio of the total surface area of a sphere and the total surface area of a hemisphere of the same radius ?

   • (A) 4 : 9
   • (B) 4 : 3 ✓
   • (C) 3 : 4
   • (D) 3 : 2

   Answer: (B)

10. 10. Bihar Board 2026

    The radii of two cylinders are in the ratio 3 : 7 and their heights are in the ratio 7 : 12, then the ratio of their curved surface areas is

    • (A) 1 : 16
    • (B) 7 : 12
    • (C) 1 : 4 ✓
    • (D) 4 : 1

    Answer: (C)

Questions parents and students ask

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

In the Bihar Board papers of 2021–2026, this chapter carried about 7 marks and about 9 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 22 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

BSEB does not publish a marking scheme. Each answer is written point by point on a structure derived from past papers and checked by a second, independent model; it is never called the board’s official scheme.