Surface Areas and Volumes: Class 9 Study Guide
Surface area is the total area of the outside of a solid. Volume is the space inside it.
The curved surface area (CSA) is the area of the curved part only. The total surface area (TSA) includes the flat faces too. Volume is measured in cubic units, such as cm³, and capacity in litres (1 litre = 1000 cm³).
In this guide, take π = 22/7 unless told otherwise.
Formulas
| Solid | Curved or total surface area | Volume |
|---|---|---|
| Cuboid (l, b, h) | TSA = 2(lb + bh + hl) | lbh |
| Cube (a) | TSA = 6a² | a³ |
| Cylinder (r, h) | CSA = 2πrh; TSA = 2πr(r + h) | πr²h |
| Cone (r, h, slant l) | CSA = πrl; TSA = πr(r + l) | ⅓πr²h |
| Sphere (r) | 4πr² | 4/3 πr³ |
| Hemisphere (r) | CSA = 2πr²; TSA = 3πr² | 2/3 πr³ |
Worked checks with r = 7 cm
A cylinder with r = 7 and h = 10: CSA = 2 × 22/7 × 7 × 10 = 440 cm², and volume = 22/7 × 49 × 10 = 1540 cm³. A cone with r = 7 and h = 24 has slant height l = √(49 + 576) = 25, so CSA = 22/7 × 7 × 25 = 550 cm² and volume = ⅓ × 22/7 × 49 × 24 = 1232 cm³. A sphere with r = 7 has surface area 4 × 22/7 × 49 = 616 cm² and volume 4/3 × 22/7 × 343 ≈ 1437.33 cm³.
And list r, h and l.
CSA, TSA or volume.
Use the same units.
Square or cubic.
Real problems
A water tank is a cuboid 2 m long, 1.5 m wide and 1 m deep. Its volume is 3 m³, which is 3000 litres, since 1 m³ = 1000 litres. A tent shaped like a cone needs fabric equal to its curved surface area, not its volume.
Worked examples and common mistakes
Example: A cube has edge 5 cm. TSA = 6 × 25 = 150 cm² and volume = 125 cm³. Common mistakes: using the height instead of the slant height for a cone's CSA, using the diameter as the radius, and mixing units such as cm and m. Halve a diameter to get the radius.
Check your understanding: model answers
1. 64 cm³. 2. 2 × 22/7 × 7 × 5 = 220 cm². 3. 4 × 22/7 × 3.5 × 3.5 = 154 cm².
✨ Key Takeaways
- CSA is the curved part only and TSA includes flat faces.
- Cylinder volume is πr²h and cone volume is ⅓πr²h.
- Sphere surface area is 4πr² and volume is 4/3 πr³.
- Use the slant height for a cone's CSA.
💬 Questions for Discussion & Reflection
- Find the volume of a cube with edge 4 cm.
- Find the CSA of a cylinder with r = 7 cm and h = 5 cm.
- A sphere has radius 3.5 cm. Find its surface area.