Surface Area, Volume and Capacity
1. Cube
A cube has 6 EQUAL square faces.
| Measurement | Formula | Example (side = 5 cm) |
|---|---|---|
| Lateral Surface Area (LSA) | 4a² | 4 × 5² = 100 cm² |
| Total Surface Area (TSA) | 6a² | 6 × 5² = 150 cm² |
| Volume | a³ | 5³ = 125 cm³ |
| Diagonal | a√3 | 5√3 ≈ 8.66 cm |
LSA = area of FOUR walls (excluding top and bottom). TSA = area of ALL six faces.
2. Cuboid
A cuboid has 6 RECTANGULAR faces (opposite faces equal).
Let length = l, breadth = b, height = h.
| Measurement | Formula | Example (5 cm × 3 cm × 4 cm) |
|---|---|---|
| LSA | 2h(l + b) | 2×4(5+3) = 64 cm² |
| TSA | 2(lb + bh + hl) | 2(15+12+20) = 94 cm² |
| Volume | l × b × h | 5×3×4 = 60 cm³ |
| Diagonal | √(l² + b² + h²) | √(25+9+16) = √50 ≈ 7.07 cm |
Worked Example: Find the TSA of a cuboid with length 8 cm, breadth 6 cm, and height 5 cm.
TSA = 2(8×6 + 6×5 + 5×8) = 2(48 + 30 + 40) = 2 × 118 = 236 cm²
3. Cylinder
A cylinder has TWO circular bases and ONE curved surface.
Let radius = r, height = h.
| Measurement | Formula | Example (r = 7 cm, h = 10 cm) |
|---|---|---|
| Curved Surface Area (CSA) | 2πrh | 2 × 22/7 × 7 × 10 = 440 cm² |
| Total Surface Area (TSA) | 2πr(r + h) | 2 × 22/7 × 7 × 17 = 748 cm² |
| Volume | πr²h | 22/7 × 49 × 10 = 1540 cm³ |
Worked Example: Find the CSA and TSA of a cylinder with radius 5 cm and height 14 cm. (Use π = 22/7)
CSA = 2 × 22/7 × 5 × 14 = 2 × 22 × 5 × 2 = 440 cm² TSA = 2 × 22/7 × 5(5 + 14) = 2 × 22/7 × 5 × 19 = 2 × 22 × 5 × 19/7 = 4180/7 ≈ 597.14 cm²
4. Hollow Cylinder
For a hollow cylinder with external radius R and internal radius r:
| Measurement | Formula |
|---|---|
| CSA (external) | 2πRh |
| CSA (internal) | 2πrh |
| Total surface area | 2πRh + 2πrh + 2π(R² — r²) |
| Volume of material | π(R² — r²)h |
5. Capacity and Volume Conversion
Capacity is the volume of liquid a container can HOLD.
Conversion: 1 m³ = 1000 litres 1 cm³ = 1 millilitre (mL) 1 litre = 1000 cm³
Worked Example: A water tank is 2 m long, 1.5 m wide, and 1 m deep. Find its capacity in litres.
Volume = 2 × 1.5 × 1 = 3 m³ Capacity = 3 × 1000 = 3000 litres
Worked Example: A cylindrical container has radius 35 cm and height 40 cm. Find its capacity in litres. (Use π = 22/7)
Volume = πr²h = 22/7 × 35² × 40 = 22/7 × 1225 × 40 = 22 × 175 × 40 = 154000 cm³ Capacity = 154000/1000 = 154 litres
6. Comparison of Solids
| Solid | TSA | Volume |
|---|---|---|
| Cube (a) | 6a² | a³ |
| Cuboid (l,b,h) | 2(lb+bh+hl) | lbh |
| Cylinder (r,h) | 2πr(r+h) | πr²h |
'For a GIVEN surface area, a sphere has the MAXIMUM volume. For a GIVEN volume, a sphere has the MINIMUM surface area.'
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'Confusing LSA and TSA' | LSA excludes the top and bottom faces. TSA includes ALL faces |
| 'Using diameter instead of radius' | ALL cylinder formulas use RADIUS (r). If diameter is given, HALVE it |
| 'Volume of cylinder = 2πrh' | That is the CSA formula. Volume = πr²h |
| 'Not converting cm³ to litres correctly' | 1 litre = 1000 cm³. Divide by 1000 to get litres |
| 'Forgetting units — cm² vs cm³' | Area: square units (cm²). Volume: cubic units (cm³) |
ICSE Exam Focus (6–8 marks)
- 2-mark questions: Find LSA/TSA of cube/cuboid given dimensions
- 3-mark questions: Find volume of cube/cuboid/cylinder
- 4-mark questions: Find missing dimension given surface area or volume
- 6-mark questions: Capacity conversion problems with cost
- 8-mark questions: Composite solids or comparison problems
Self-Test
Q1. Find the TSA of a cube with side 8 cm. A1. TSA = 6 × 8² = 6 × 64 = 384 cm².
Q2. Find the volume of a cuboid with l = 12 cm, b = 8 cm, h = 5 cm. A2. Volume = 12 × 8 × 5 = 480 cm³.
Q3. Find the CSA of a cylinder with radius 7 cm and height 12 cm. (π = 22/7) A3. CSA = 2 × 22/7 × 7 × 12 = 2 × 22 × 12 = 528 cm².
Q4. A water tank is 3 m × 2 m × 1.5 m. Find its capacity in litres. A4. Volume = 3 × 2 × 1.5 = 9 m³. Capacity = 9 × 1000 = 9000 litres.
Q5. The TSA of a cube is 294 cm². Find its side and volume. A5. 6a² = 294 → a² = 49 → a = 7 cm. Volume = 7³ = 343 cm³.
Q6. A cylindrical tank has radius 1.4 m and height 2 m. Find its capacity in litres. (π = 22/7) A6. Volume = 22/7 × 1.4² × 2 = 22/7 × 1.96 × 2 = 22 × 0.28 × 2 = 12.32 m³. Capacity = 12320 litres.
