Surface Areas and Volumes — Class 10 Mathematics
"From cricket ball to ice-cream cone, from water tank to pencil — every shape can be measured."
1. About the Chapter
This chapter calculates surface areas and volumes of 3D shapes:
- Cube, Cuboid
- Cylinder
- Cone
- Sphere, Hemisphere
- Frustum (cone with top cut off)
- Combinations of these
- Conversions between shapes
Why Important
- Engineering (tanks, pipes, machines)
- Architecture (buildings, domes)
- Daily life (containers, packaging)
- Industrial (storage, transport)
2. Recap — Basic 3D Shapes
Cube (side a)
- Surface Area = 6a²
- Volume = a³
Cuboid (L × B × H)
- Surface Area = 2(LB + BH + HL)
- Volume = L × B × H
Cylinder (radius r, height h)
- Curved Surface Area (CSA) = 2πrh
- Total Surface Area (TSA) = 2πr(r + h)
- Volume = πr²h
Cone (radius r, height h, slant l)
- Slant height l = √(r² + h²)
- CSA = πrl
- TSA = πr(l + r)
- Volume = (1/3)πr²h
Sphere (radius r)
- Surface Area = 4πr²
- Volume = (4/3)πr³
Hemisphere (radius r)
- Curved Surface Area = 2πr²
- Total Surface Area = 3πr² (curved + flat circle)
- Volume = (2/3)πr³
Frustum (radius R, r at top and bottom, height h, slant l)
- Slant l = √(h² + (R−r)²)
- CSA = π(R + r)l
- TSA = π[(R + r)l + R² + r²]
- Volume = (1/3)πh(R² + r² + Rr)
3. Combination of Solids
Strategy
Many real objects combine 2+ shapes:
- Ice-cream cone (cone + hemisphere)
- Capsule (cylinder + 2 hemispheres)
- Building (cuboid + half-cylinder)
- Pencil (cylinder + cone)
Procedure
- Identify the simple shapes
- Calculate each part separately
- ADD or SUBTRACT appropriately
Example: Ice-Cream Cone
A cone with radius 5 cm and height 12 cm, topped by hemisphere of radius 5 cm.
Surface Area (assuming flat base of cone exposed):
- Cone slant = √(25+144) = 13 cm
- Cone CSA = π × 5 × 13 = 65π
- Hemisphere CSA = 2π(5)² = 50π
- Total (cone CSA + hemisphere CSA) = 115π ≈ 361 cm²
Volume (full cone + hemisphere):
- Cone: (1/3)π(25)(12) = 100π
- Hemisphere: (2/3)π(125) = 250π/3
- Total: 100π + 250π/3 = 550π/3 ≈ 575.5 cm³
Example: Capsule
A medicine capsule is a cylinder of radius 5 mm and length 14 mm, with hemispheres at both ends.
Total length = 14 + 2(5) = 24 mm Volume:
- Cylinder: π(25)(14) = 350π
- 2 hemispheres = 1 sphere = (4/3)π(125) = 500π/3
- Total: 350π + 500π/3 = 1550π/3 ≈ 1623 mm³
4. Conversion of Solids
Concept
When one solid is recast into another, volume is conserved.
Volume of original = Volume of new shape.
Examples
Example 1: A metal sphere of radius 5 cm is melted and recast into a cylinder of radius 2 cm. Find the height of the cylinder.
- Volume of sphere = (4/3)π(125) = 500π/3
- Volume of cylinder = π(4)h
- Set equal: 4πh = 500π/3
- h = 500/12 = 41.67 cm
Example 2: How many lead shots of diameter 1 cm can be made from a lead cuboid 1 m × 50 cm × 25 cm?
- Cuboid volume = 100 × 50 × 25 = 125,000 cm³
- Each lead shot (sphere) radius = 0.5 cm
- Volume of each shot = (4/3)π(0.5)³ = (4/3)π(0.125) = π/6 cm³
- Number of shots = 125,000 / (π/6) = 750,000/π ≈ 238,732
Example 3: Cube of side 6 cm melted to form 4 cuboids. If cuboid has 3 cm × 2 cm base, find height of cuboid.
- Cube volume = 216 cm³
- 4 cuboids = 216 → each cuboid volume = 54 cm³
- Cuboid volume = 3 × 2 × h = 6h
- 6h = 54 → h = 9 cm
5. Frustum of a Cone
What is a Frustum?
When a cone is cut PARALLEL to its base, the remaining part below the cut is a frustum (also called truncated cone).
Used For
- Buckets, drinking glasses, lamp shades
- Industrial containers
Formulas
For a frustum with radii R (lower) and r (upper), height h:
Slant height (l): l = √(h² + (R − r)²)
CSA: CSA = π(R + r)l
TSA: TSA = π[(R + r)l + R² + r²]
Volume: V = (1/3)πh(R² + r² + Rr)
Example
A bucket has radii 25 cm (lower) and 15 cm (upper), height 30 cm.
- Slant l = √(900 + 100) = √1000 ≈ 31.6 cm
- Volume = (1/3)π(30)(625 + 225 + 375) = 10π × 1225 = 12250π ≈ 38,500 cm³
6. Worked Examples
Example 1: Cylinder and Cone
A cylinder of height 14 cm and radius 7 cm has a cone of same dimensions on top. Find total volume.
- Cylinder: πr²h = (22/7) × 49 × 14 = 2156 cm³
- Cone: (1/3)πr²h_cone — but height of cone needed. Assume same height = 14.
- Cone: (1/3) × (22/7) × 49 × 14 = 2156/3 ≈ 718.67 cm³
- Total: 2156 + 718.67 ≈ 2874.67 cm³
Example 2: Hemisphere on Cylinder
A solid is a cylinder of height 8 cm, radius 5 cm, with hemispheres at each end. Find total surface area.
- Length of cylinder + 2 hemispheres (cylindrical part): 8 cm
- TSA = CSA of cylinder + 2 × CSA of hemisphere
- = 2πr × 8 + 2 × 2πr²
- = 2π × 5 × 8 + 4π × 25
- = 80π + 100π = 180π
- ≈ 565.5 cm²
Example 3: Hollow Cylinder
A hollow cylinder has outer radius 10 cm, inner radius 8 cm, height 15 cm. Find volume of material.
- Outer cylinder volume = π(100)(15) = 1500π
- Inner cylinder volume = π(64)(15) = 960π
- Material volume = 1500π − 960π = 540π ≈ 1696.5 cm³
7. Common Mistakes
-
Wrong formula confusion
- Cone volume has (1/3) factor; cylinder doesn't.
- Sphere has (4/3) factor.
-
Slant height vs height
- Slant height: along the slope (cone, frustum)
- Vertical height: straight up
-
CSA vs TSA
- CSA = curved only (no top/bottom)
- TSA = curved + flat ends
-
Hemisphere TSA
- Hemisphere TSA = 3πr² (curved 2πr² + flat πr²)
- Not 4πr² (that's full sphere)
-
Forgetting to add/subtract
- In combinations: think carefully which surfaces are exposed.
8. Real-World Applications
Engineering
- Water tank (cylinder)
- Storage silos (cone + cylinder)
- Pipes (hollow cylinder)
- Capsule design
Architecture
- Domes (hemispheres)
- Spires (cones)
- Indian temple gopuram (pyramid + cone-like)
Manufacturing
- Bottle and can design
- Packaging optimisation
- Material costs
Daily Life
- Ice cream cones
- Tents (cone, dome)
- Water tanks
- Cylindrical containers
9. Indian Context
Ancient Indian Measurements
- Sulba Sutras gave precise volume formulas
- Used for fire-altar construction
- Aryabhata computed volumes accurately
Modern Indian
- Water tank manufacturing (Vajra, Sintex)
- LPG cylinder manufacturing
- ISRO rocket design uses precise volume calculations
10. Conclusion
Surface Areas and Volumes connect geometry to real-world objects:
- Cube to cylinder to cone to sphere — basic shapes
- Combinations reflect real objects
- Conversions are everywhere in industry
Master:
- ALL formulas for 6 basic shapes + frustum
- COMBINATION strategies
- CONVERSION principle (volume conserved)
Practice 20+ problems. This is HIGH-MARK chapter for board exam.
Every 3D shape can be measured. This chapter gives you the tools.
