Cylinder, Cone and Sphere (Surface Area and Volume)

Q: A closed cylinder has radius 3.5 cm and height 10 cm. Find its curved surface area, total surface area and volume.

2π r h = 220 cm2; 2π r(h + r) = 22 × 13.5 = 297 cm2; π r2 h = 385 cm3.

Q: A cone has radius 6 cm and height 8 cm. Find its curved surface area and volume, using π = 3.14.

l = 10 cm; curved surface area = π r l = 60π = 188.4 cm2; volume = (1)/(3)π × 36 × 8 = 96π = 301.44 cm3.

Q: A sphere has diameter 21 cm. Find its surface area and volume.

r = 10.5 cm; surface area = 1386 cm2; volume = 4851 cm3.

Q: The radius of a sphere is tripled. By what factor do its surface area and its volume change?

Surface area is multiplied by 32 = 9; volume is multiplied by 33 = 27.

Q: A hollow cylinder is 14 cm long, with outer radius 5 cm and inner radius 4 cm. Find the volume of material in it.

π (R2 - r2) h = (22)/(7) × 9 × 14 = 396 cm3.

Q: A solid cone of radius 6 cm and height 24 cm is melted and recast into a sphere. Find the radius of the sphere.

(1)/(3)π × 36 × 24 = (4)/(3)π R3 gives R3 = 216, so R = 6 cm.

ICSE Class 10 · Mathematics · Chapter 20

What this chapter covers

Solids, surfaces and space
Right circular cylinder: curved and total surface area
Right circular cylinder: volume
Hollow cylinders: inner and outer measurements
Right circular cone: slant height and surface area
Right circular cone: volume
Sphere and hemisphere: surface area
Sphere and hemisphere: volume
Direct application problems: cost of materials and work
Melting and recasting
Combination of solids

Key definitions

Surface area is the total area of all the surfaces that enclose a solid, measured in square units such as cm² or m².

Curved surface area is the area of only the curved part of a solid's surface, leaving out any flat faces.

Total surface area is the sum of the curved surface area and the areas of all the flat faces of a solid.

Volume is the amount of three-dimensional space a solid occupies, measured in cubic units such as cm³ or m³.

Capacity is the volume of liquid or other material that a hollow container can hold, usually stated in litres or millilitres.

Key points

Curved surface area counts only the curved part; total surface area adds every flat face that is present.
Cylinder: curved surface 2π r h, total surface 2π r(h + r), volume π r² h.
Hollow cylinder: material π (R² - r²) h; inner volume π r² h and outer volume π R² h.
Cone: l = √(r² + h²), curved surface π r l, total surface π r(l + r), volume (1)/(3)π r² h with vertical height h.
Sphere: surface 4π r², volume (4)/(3)π r³; spherical shell (4)/(3)π (R³ - r³).
Hemisphere: curved surface 2π r², total surface of a solid hemisphere 3π r², volume (2)/(3)π r³.
Painting, plastering and covering use area; digging, filling and quantity of material use volume; match units to the rate before multiplying.
In melting and recasting, volume is conserved but surface area usually changes.
For combined solids, add or subtract volumes, but count only the surfaces that are exposed.
1 m³ = 1000 litres, 1 litre = 1000 cm³, and 1 m² = 10 000 cm².

Questions answered

Define a right circular cone and state the relation between its radius, height and slant height.

A right circular cone is a solid with a circular base and a vertex, where the line from the vertex to the centre of the base is perpendicular to the base. The relation is l² = r² + h².

A closed cylinder has radius 3.5 cm and height 10 cm. Find its curved surface area, total surface area and volume.

2π r h = 220 cm²; 2π r(h + r) = 22 × 13.5 = 297 cm²; π r² h = 385 cm³.

A cone has radius 6 cm and height 8 cm. Find its curved surface area and volume, using π = 3.14.

l = 10 cm; curved surface area = π r l = 60π = 188.4 cm²; volume = (1)/(3)π × 36 × 8 = 96π = 301.44 cm³.

A sphere has diameter 21 cm. Find its surface area and volume.

r = 10.5 cm; surface area = 1386 cm²; volume = 4851 cm³.

Why is the volume of a cone one-third of the volume of a cylinder with the same base and height?

A cone tapers to a point, so it holds much less than the cylinder; filling the cylinder takes exactly three cone-fuls, so V_cone = (1)/(3)π r² h while V_cylinder = π r² h.

The radius of a sphere is tripled. By what factor do its surface area and its volume change?

Surface area is multiplied by 3² = 9; volume is multiplied by 3³ = 27.

A hollow cylinder is 14 cm long, with outer radius 5 cm and inner radius 4 cm. Find the volume of material in it.

π (R² - r²) h = (22)/(7) × 9 × 14 = 396 cm³.

A solid cone of radius 6 cm and height 24 cm is melted and recast into a sphere. Find the radius of the sphere.

(1)/(3)π × 36 × 24 = (4)/(3)π R³ gives R³ = 216, so R = 6 cm.

Past board exam questions

ICSE 2027 sample paper · 1 mark

A conical tent with a capacity of 616 m³ has a base area 154 m². The height of the tent is:

3 m
4 m
7 m
12 m

Show answer

12 m

ICSE 2027 sample paper · 4 marks

Three students A, B and C of Class 10 are each given an equal amount of Play- Dough to form different shaped solids for a mathematics project. A forms a cone of radius 7 cm and height 28 cm. B forms a sphere of radius r and C forms a cylinder of radius 14 cm. Find the: (a) radius r of the sphere so formed. (b) height of the cylinder so formed. (c) Volume of the dough given to each of them. Give your answers to (b) and (c) correct to the nearest whole number. (Use π = (22)/(7))

Show answer

(a) Volume of sphere = volume of cone: (4)/(3)π r³ = (1)/(3)π (7)² (28) ⇒ r³ = (49 × 28)/(4) = 343 ⇒ r = 7 cm. (b) Volume of cylinder = volume of sphere: π (14)² H = (4)/(3)π (7)³ ⇒ H = (4 × 343)/(3 × 196) = (7)/(3) = 2.33 cm ≈ 2 cm. (c) Volume of dough = (1)/(3) × (22)/(7) × 7² × 28 = (4312)/(3) = 1437.33 cm³ ≈ 1437 cm³.

ICSE 2027 sample paper · 4 marks

Two beakers (drinking glasses) are cylindrical in shape having diameter and height 7 cm and 10 cm respectively. Glass A has hemispherical raised bottom, and glass B has a conical raised bottom of height 3.5 cm. Determine which glass has a greater capacity and calculate the difference in their capacities. Give your answer correct to 2 decimal places. (Use π = (22)/(7))

Show answer

r = 3.5 cm, h = 10 cm. V_A = π r² h - (2)/(3)π r³, V_B = π r² h - (1)/(3)π r² (3.5) = π r² h - (1)/(3)π r³ (since the cone's height = r = 3.5 cm). So V_B > V_A: glass B has the greater capacity. V_B - V_A = (2)/(3)π r³ - (1)/(3)π r³ = (1)/(3) × (22)/(7) × 3.5³ = 44.916… Difference = 44.92 cm³ (V_A = 295.17 cm³, V_B = 340.08 cm³).

ICSE 2026 sample paper · 1 mark

A solid sphere is cut into two identical hemispheres. Assertion (A): The total volume of two hemispheres is equal to the volume of the original sphere. Reason (R): The total surface area of two hemispheres together is equal to the surface area of the original sphere.

(A) is true, (R) is false.
(A) is false, (R) is true.
Both (A) and (R) are true and (R) is the correct explanation of (A).
Both (A) and (R) are true, but (R) is not the correct explanation of (A).

Show answer

(A) is true, (R) is false.

ICSE 2026 board paper · 1 mark

The total surface area of a solid sphere (S₁) of radius R and a solid hemisphere (S₂) of radius r are equal. The ratio of radii R and r is:

1 : 1
2 : 1
√(3) : 2
2 : √(3)

Show answer

√(3) : 2

ICSE 2026 sample paper · 4 marks

A manufacturing company prepares spherical ball bearings, each of radius 7 mm and mass 4 gm. These ball bearings are packed into boxes. Each box can have a maximum of 2156 cm³ of ball bearings. Find the: (a) maximum number of ball bearings that each box can have. (b) mass of each box of ball bearings in kg. (Use π = (22)/(7))

Show answer

Radius = 7 mm = (7)/(10) cm. Volume of one ball bearing = (4)/(3) × (22)/(7) × ((7)/(10))³ = (4312)/(3000) cm³. (a) Number of ball bearings = (2156)/((4)/(3) × (22)/(7) × (343)/(1000)) = (2156 × 3 × 7 × 1000)/(4 × 22 × 343) = 1500. (b) Mass = 1500 × 4 g = 6000 g = 6 kg.

ICSE 2025 compartment paper · 1 mark

The ratio of diameters of two right circular cones is 3 : 7 and that of their heights is 14 : 9, then their volumes are in ratio:

3 : 7
2 : 7
3 : 2
9 : 49

Show answer

2 : 7

ICSE 2025 sample paper · 1 mark

A solid sphere is cut into two identical hemispheres. Statement 1: The total volume of two hemispheres is equal to the volume of the original sphere. Statement 2: The total surface area of two hemispheres together is equal to the surface area of the original sphere. Which of the following is valid?

Both the statements are true.
Both the statements are false.
Statement 1 is true, and Statement 2 is false.
Statement 1 is false, and Statement 2 is true.

Show answer

Statement 1 is true, and Statement 2 is false.