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Class 9 Maths • Chapter 11 • New NCERT Syllabus

Surface Areas and Volumes

Vardaan Learning Institute  |  Right Circular Cone, Sphere, Hemisphere

Syllabus Update Note In the latest rationalised syllabus, formulas related to Cuboid, Cube, and Cylinder have been removed from the Class 9 curriculum. Only Cone, Sphere, and Hemisphere are in scope. Keep $\pi = \frac{22}{7}$ unless stated otherwise.

📐 1. Master Formula Sheet

1. Right Circular Cone
h r l
Slant Height (l) √(r² + h²)
Curved Surface Area (CSA) πrl
Total Surface Area (TSA) πr(l + r)
Volume (V) ⅓πr²h
2. Sphere
r
Surface Area (SA) 4πr²
Volume (V) 4/3 πr³
*Sphere only has one surface area, no distinct CSA/TSA.
3. Solid Hemisphere
r r
Curved Surface Area (CSA) 2πr²
Total Surface Area (TSA) 3πr²
Volume (V) 2/3 πr³

⚠️ 2. Common Pitfalls & Conversions

• Radius vs Diameter: Always check if the question gives diameter! If $d=14$, immediately write $r=7$.
• Canvas for a Tent: Making a conical tent involves ONLY the Curved Surface Area (CSA), not the base. Base is the ground.
• Hollow Hemisphere Bowl: The inner and outer curved surfaces are calculated separately. Inner TSA of bowl = Inner CSA + top ring area (if thick).
Crucial Volume Conversions
1 $cm^3$ = 1 millilitre (mL)
1000 $cm^3$ = 1 Litre (L)
1 $m^3$ = 1000 Litres (L) = 1 kilolitre (kL)
✏️ Practice Questions — Surface Areas & Volumes (30 Questions)

Section A — 1 Mark Direct Formula Base Easy

Q1. Find slant height of cone with r=3, h=4.
Q2. Find TSA of a solid hemisphere of radius 10cm. (Use $\pi=3.14$).
Q3. The radius of a sphere is 2r. What is its volume?
Q4. Surface area of a sphere of radius 7cm?
Q5. Area of the base of a cone is $154 cm^2$. Find radius.
Q6. Find CSA of cone whose base radius is 7cm and slant height is 10cm.
Q7. Volume of a cone is $\frac{1}{3}\pi r^2 h$. If height is doubled, volume becomes?
Q8. Difference between TSA and CSA of a solid hemisphere of radius r?

Section B — Core Word Problems Medium

Q9. A conical tent is 10m high and base radius 24m. Find slant height and cost of canvas at ₹70 per $m^2$.
Q10. Find the radius of a sphere whose surface area is $154 cm^2$.
Q11. The radius of a spherical balloon increases from 7cm to 14cm as air is pumped. Find ratio of surface areas in the two cases.
Q12. A hemispherical bowl made of brass has inner diameter 10.5cm. Find cost of tin-plating it on the inside at ₹16 per $100 cm^2$.
Q13. Find the volume of a right circular cone with radius 6cm, height 7cm.
Q14. The capacity of a conical vessel with height 12cm and slant height 13cm in litres?
Q15. A capsule of medicine is in the shape of a sphere of diameter 3.5mm. How much medicine ($mm^3$) is needed to fill it?

Section C — NCERT Deep Dive Hard

  1. Q16. What length of tarpaulin 3m wide will be required to make a conical tent of height 8m and base radius 6m? Assume extra length for margins is 20cm. (Use $\pi=3.14$).
  2. Q17. A joker's cap is in the form of a right circular cone of base radius 7cm and height 24cm. Find area of sheet required to make 10 such caps.
  3. Q18. A hemispherical bowl is made of steel, 0.25cm thick. The inner radius of the bowl is 5cm. Find the outer curved surface area of the bowl.
  4. Q19. A right circular cylinder just encloses a sphere of radius r. Find (i) surface area of sphere, (ii) CSA of cylinder, (iii) ratio of the areas obtained. (Standard NCERT proof).
  5. Q20. A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of ₹4989.60. If cost is ₹20 per sqm, find (i) inside SA, (ii) volume of air inside the dome.
  6. Q21. Twenty-seven solid iron spheres, each of radius r and surface area S, are melted to form a sphere with surface area $S'$. Find (i) radius $r'$ of new sphere, (ii) ratio of S to $S'$.

Section D — Competency & Board Level Hard

  1. Q22. A right triangle ABC with sides 5cm, 12cm and 13cm is revolved about the side 12cm. Find the volume of the solid so obtained. (Forms a cone).
  2. Q23. If the same triangle in Q22 is revolved about the side 5cm, find the volume. Find the ratio of volumes in Q22 and Q23.
  3. Q24. Assuming the earth to be a sphere, and the moon also. The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?
  4. Q25. How many litres of milk can a hemispherical bowl of diameter 10.5cm hold?
  5. Q26. Water flows at the rate of 10m/minute through a cylindrical pipe 5mm in diameter. How long would it take to fill a conical vessel whose diameter at the base is 40cm and depth 24cm? (Wait: Cylinder pipe concept might be outside syllabus, try applying volume equalization).
  6. Q27. The radius and slant height of a cone are in ratio 4:7. If its CSA is $792cm^2$, find its radius.
  7. Q28. The internal and external diameters of a hollow hemispherical vessel are 24cm and 25cm. The cost of painting $1cm^2$ surface is ₹0.05. Find total cost. (Inner CSA + Outer CSA + Ring Area).
  8. Q29. Radius of a sphere is increased by 10%. Prove that its volume will increase by 33.1%.
  9. Q30. A wooden bookshelf has external dimensions 85x110x25cm, thickness 5cm. Out of syllabus (Cuboid), skip or replace: A spherical balloon's volume increases from $V_1$ to $V_2$ when radius doubles, find $V_1 : V_2$. (Ans: 1:8).
✅ Key Checkpoints: Q1: l=5 | Q2: $3 \times 3.14 \times 100 = 942$ | Q3: $\frac{32}{3}\pi r^3$ | Q8: $\pi r^2$ (area of circular base) | Q9: l=26m, CSA=1961.14 $m^2$, Cost=₹137280. | Q11: 1:4 | Q16: l=10, CSA=188.4, length=188.4/3 = 62.8m + 0.2 = 63m. | Q19: 1:1 ratio. | Q22: Cone with r=5, h=12. V=$100\pi$. | Q23: Cone with r=12, h=5. V=$240\pi$. Ratio = 5:12. | Q24: 1/64.