Math of Space: Surface Area and Volume – Class 9 Mathematics notes, important questions and MCQs

Chapter overview

Surface area is the total area covering a solid; volume is the space it fills. Each common solid has its own formulas.

Cuboid and cube

• Cuboid: TSA = 2(lb + bh + hl), lateral area = 2h(l + b), V = lbh.
• Cube: TSA = 6a², lateral area = 4a², V = a³.

Round solids

• Cylinder: CSA = 2πrh, TSA = 2πr(r + h), V = πr²h.
• Cone: CSA = πrl, TSA = πr(l + r), V = ⅓πr²h, l = √(r² + h²).
• Sphere: SA = 4πr², V = (4/3)πr³. Hemisphere: CSA = 2πr², TSA = 3πr², V = (2/3)πr³.

Key terms

Lateral surface area

area of the side faces only

Capacity

volume a container can hold (1 L = 1000 cm³)

Slant height

distance from the cone's vertex to the edge of its base

Formulas

• Cylinder V = πr²h
• Cone V = ⅓πr²h
• Sphere V = (4/3)πr³
• 1 m³ = 1000 L

Important questions with answers

Quick revision

• Cone volume is one-third of the cylinder with the same base and height.
• Hemisphere TSA = 3πr².
• 1000 cm³ = 1 litre.

Exam tips

• For open containers (no lid), leave out one face.
• If radius doubles, a sphere's volume becomes 8 times.

Practice MCQs

1. Volume of a cuboid 5 × 4 × 3 cm is

   1. 60 cm³
   2. 12 cm³
   3. 47 cm³
   4. 120 cm³
   Show answer

   Answer: (a) 60 cm³. 5 × 4 × 3.

2. Total surface area of a cube of side 3 cm is

   1. 27 cm²
   2. 54 cm²
   3. 36 cm²
   4. 18 cm²
   Show answer

   Answer: (b) 54 cm². 6 × 9.

3. 1 litre equals

   1. 100 cm³
   2. 10 cm³
   3. 1000 cm³
   4. 1 m³
   Show answer

   Answer: (c) 1000 cm³. 1000 cm³.

4. CSA of a cylinder with r = 7 cm, h = 10 cm is

   1. 1540 cm²
   2. 220 cm²
   3. 440 cm²
   4. 154 cm²
   Show answer

   Answer: (c) 440 cm². 2 × 22/7 × 7 × 10 = 440.

5. Volume of a cone with r = 3 cm and h = 7 cm is

   1. 22 cm³
   2. 198 cm³
   3. 66 cm³
   4. 132 cm³
   Show answer

   Answer: (c) 66 cm³. ⅓ × 22/7 × 9 × 7 = 66.

6. Surface area of a sphere of radius 7 cm is

   1. 154 cm²
   2. 616 cm²
   3. 308 cm²
   4. 1437 cm²
   Show answer

   Answer: (b) 616 cm². 4 × 22/7 × 49 = 616.

7. If the radius of a sphere is doubled, its surface area becomes

   1. 2 times
   2. 4 times
   3. 8 times
   4. 16 times
   Show answer

   Answer: (b) 4 times. SA ∝ r².

8. A conical tent has radius 7 m and height 24 m. Canvas needed for its curved surface is

   1. 528 m²
   2. 550 m²
   3. 1232 m²
   4. 616 m²
   Show answer

   Answer: (b) 550 m². l = 25; πrl = 22/7 × 7 × 25 = 550.