Surface Areas and Volumes – Class 10 Mathematics notes, important questions and MCQs

Chapter overview

Solids like a toy (cone on a hemisphere) or a capsule (cylinder with two hemispheres) are combinations of basic solids. Add only the surfaces that are visible; volumes simply add up.

Basic solids

• Cuboid: V = lbh; TSA = 2(lb + bh + hl).
• Cylinder: V = πr²h; CSA = 2πrh.
• Cone: V = ⅓πr²h; CSA = πrl with l = √(r² + h²).
• Sphere: V = (4/3)πr³; SA = 4πr². Hemisphere: V = (2/3)πr³; CSA = 2πr².

Combinations

• Surface area of a combined solid = sum of the outer (exposed) surfaces only.
• Volume of a combined solid = sum of volumes of the parts.
• When one solid is melted and recast, the volume stays the same.

Key terms

Curved surface area

area of the curved part only

Total surface area

curved area plus the flat faces

Slant height

l = √(r² + h²) of a cone

Formulas

• Cone V = ⅓πr²h, CSA = πrl
• Sphere V = (4/3)πr³, SA = 4πr²
• Hemisphere V = (2/3)πr³, TSA = 3πr²
• Cylinder V = πr²h, TSA = 2πr(r + h)

Important questions with answers

Quick revision

• Visible surfaces add for area; all parts add for volume.
• Volume is conserved on melting.
• l² = r² + h² for a cone.

Exam tips

• Do not add the hidden circular face where two solids join.
• For melting/recasting questions, equate volumes and keep the units the same.

Practice MCQs

1. Volume of a cube of side 4 cm is

   1. 16 cm³
   2. 64 cm³
   3. 48 cm³
   4. 96 cm³
   Show answer

   Answer: (b) 64 cm³. 4³ = 64 cm³.

2. Curved surface area of a hemisphere of radius r is

   1. πr²
   2. 2πr²
   3. 3πr²
   4. 4πr²
   Show answer

   Answer: (b) 2πr². Half of the sphere's 4πr².

3. Slant height of a cone with r = 6 cm, h = 8 cm is

   1. 14 cm
   2. 10 cm
   3. 7 cm
   4. 12 cm
   Show answer

   Answer: (b) 10 cm. √(36 + 64) = 10 cm.

4. Two cubes of side 5 cm are joined end to end. TSA of the cuboid is

   1. 250 cm²
   2. 300 cm²
   3. 200 cm²
   4. 150 cm²
   Show answer

   Answer: (a) 250 cm². Cuboid 10 × 5 × 5: 2(50 + 25 + 50) = 250 cm².

5. A cone and a cylinder have the same base and height. Their volumes are in the ratio

   1. 1 : 3
   2. 3 : 1
   3. 1 : 2
   4. 2 : 3
   Show answer

   Answer: (a) 1 : 3. ⅓πr²h : πr²h.

6. Volume of a sphere of radius 3 cm (in terms of π) is

   1. 36π cm³
   2. 27π cm³
   3. 12π cm³
   4. 108π cm³
   Show answer

   Answer: (a) 36π cm³. (4/3)π × 27 = 36π.

7. A metal sphere of radius 6 cm is melted into a wire of radius 0.2 cm. The length of the wire is

   1. 36 m
   2. 7.2 m
   3. 72 m
   4. 720 m
   Show answer

   Answer: (c) 72 m. (4/3)π × 216 = π × 0.04 × h ⇒ h = 7200 cm = 72 m.

8. A toy is a cone of radius 3.5 cm on a hemisphere of the same radius; total height is 15.5 cm. Its total surface area is

   1. 291.5 cm²
   2. 154 cm²
   3. 77 cm²
   4. 214.5 cm²
   Show answer

   Answer: (d) 214.5 cm². h_cone = 12, l = 12.5; πrl + 2πr² = 137.5 + 77 = 214.5 cm².