Surface Areas and Volumes
This Class 9 chapter gives the surface area and volume formulas for the common solids — the cuboid, cube, cylinder, cone, sphere and hemisphere. The work is mostly choosing the right formula and substituting carefully. It is formula-heavy but highly scoring, and it sets up the combination-of-solids problems of Class 10.
Learning objectives
- Recall surface area and volume formulas for standard solids.
- Find the surface area and volume of a cuboid and a cube.
- Find the curved/total surface area and volume of a cylinder and a cone.
- Find the surface area and volume of a sphere and a hemisphere.
- Apply the formulas to word problems.
Key concepts
Cuboid and cube
For a cuboid with length l, breadth b and height h: total surface area = 2(lb + bh + hl) and volume = l × b × h. A cube of side a is a special cuboid: surface area = 6a², volume = a³.
Cylinder
For a cylinder of radius r and height h: curved surface area = 2πrh, total surface area = 2πr(r + h), and volume = πr²h.
Cone
For a cone of radius r, height h and slant height l = √(r² + h²): curved surface area = πrl, total surface area = πr(r + l), and volume = ⅓πr²h.
Sphere and hemisphere
For a sphere of radius r: surface area = 4πr², volume = (4/3)πr³. For a hemisphere: curved surface area = 2πr², total surface area = 3πr², volume = (2/3)πr³.
Important formulas
Cuboid
TSA = 2(lb + bh + hl), V = lbh
Cube
TSA = 6a², V = a³
Cylinder
CSA = 2πrh, TSA = 2πr(r + h), V = πr²h
Cone
CSA = πrl, V = ⅓πr²h, l = √(r² + h²)
Sphere / hemisphere
Sphere: 4πr², (4/3)πr³ ; Hemisphere: 3πr², (2/3)πr³
Key definitions
- Surface area
- The total area of the outer surfaces of a solid.
- Volume
- The amount of space occupied by a solid.
- Slant height
- The distance from the apex of a cone to the edge of its base, l = √(r² + h²).
- Cuboid
- A box-shaped solid with six rectangular faces.
Solved examples
Q1. Find the volume of a cube of side 4 cm.
Solution: Volume = a³ = 4³ = 64 cm³.
Q2. Find the volume of a cuboid measuring 5 cm × 4 cm × 3 cm.
Solution: Volume = l × b × h = 5 × 4 × 3 = 60 cm³.
Q3. Find the surface area of a sphere of radius 7 cm. (π = 22/7.)
Solution: Surface area = 4πr² = 4 × (22/7) × 49 = 616 cm².
Common mistakes to avoid
- Confusing total surface area with curved surface area.
- Using the diameter in place of the radius.
- Forgetting to find the slant height before a cone's CSA.
- Mixing up the sphere (4/3 πr³) and hemisphere (2/3 πr³) volumes.
Surface Areas and Volumes — MCQ Quiz
13 questions with instant feedback. Use number keys 1–4 to answer.
The volume of a cube of side a is:
Practice questions
Short answer
Write the volume formula of a cone.
V = ⅓πr²h.
Find the surface area of a cube of side 5 cm.
6 × 5² = 150 cm².
What is the total surface area of a cylinder?
2πr(r + h).
Long answer
A closed cylindrical tank has radius 7 m and height 10 m. Find its total surface area. (π = 22/7)
TSA = 2πr(r + h) = 2 × (22/7) × 7 × (7 + 10) = 2 × 22 × 17 = 748 m².
Find the volume and surface area of a sphere of radius 3 cm (leave answers in terms of π).
Volume = (4/3)π(3)³ = (4/3)π × 27 = 36π cm³. Surface area = 4π(3)² = 36π cm².
HOTS (Higher Order Thinking)
A solid cube of side 6 cm is cut into smaller cubes of side 2 cm. How many small cubes are formed?
Number = (6/2)³ = 3³ = 27 small cubes (volume 216 ÷ 8 = 27).
If the radius of a sphere is doubled, how does its volume change?
Volume depends on r³, so doubling the radius multiplies the volume by 2³ = 8 times.
Quick revision
Revision notes
- Cuboid: TSA 2(lb+bh+hl), V lbh; Cube: 6a², a³.
- Cylinder: CSA 2πrh, TSA 2πr(r+h), V πr²h.
- Cone: CSA πrl, V ⅓πr²h, l = √(r²+h²).
- Sphere: 4πr², (4/3)πr³; Hemisphere: 3πr², (2/3)πr³.
Key takeaways
- Keep the formula table at your fingertips.
- Find the slant height first for cone surface areas.
- Always substitute the radius, not the diameter.
Frequently asked questions
What solids are covered in Class 9?
Cuboid, cube, cylinder, cone, sphere and hemisphere.
What is the difference between CSA and TSA?
CSA (curved surface area) excludes the flat faces; TSA (total surface area) includes them.
How is the slant height of a cone found?
Using l = √(r² + h²), from the radius and the vertical height.