Solids — Surface Area and Volume
Overview
Surface area and volume of three-dimensional solids form a core component of the WB TET Mathematics section. This topic tests your ability to visualise shapes, apply correct formulas, and perform accurate calculations under time pressure. At the primary and upper-primary level, the focus remains on four fundamental solids: cube, cuboid, cylinder, and sphere.
Understanding these concepts is not just about memorising formulas. You must grasp what surface area represents (total outer covering of a solid) versus what volume represents (space enclosed within). Exam questions typically ask you to find one quantity given dimensions, compare solids, or solve word problems involving real-life objects like water tanks, boxes, or balls. Mastery here also supports pedagogy questions where you may need to explain how children develop spatial reasoning.
Expect 2–4 direct questions on this topic. Speed and accuracy depend on formula recall and careful unit handling.
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Key Concepts
- **Surface Area** is the total area of all outer faces of a solid. Measured in square units (cm², m²).
- **Volume** is the amount of three-dimensional space a solid occupies. Measured in cubic units (cm³, m³, litres where 1 litre = 1000 cm³).
- **Lateral Surface Area (LSA)** or **Curved Surface Area (CSA)** excludes the top and bottom faces — only the "side" surface.
- **Total Surface Area (TSA)** includes all faces: lateral/curved surface plus the base(s).
- A **cube** has all edges equal; a **cuboid** has three distinct pairs of equal edges (length, breadth, height).
- A **cylinder** has two identical circular bases connected by a curved surface.
- A **sphere** has no edges or vertices — every point on its surface is equidistant from the centre.
- When a solid is hollow (like a pipe), you may need to calculate inner and outer surface areas or the volume of material used.
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Formulas / Key Facts
### Cube (edge = a) | Quantity | Formula | |----------|---------| | Volume | a³ | | LSA (4 faces) | 4a² | | TSA (6 faces) | 6a² | | Diagonal of face | a√2 | | Space diagonal | a√3 |
### Cuboid (length = l, breadth = b, height = h) | Quantity | Formula | |----------|---------| | Volume | l × b × h | | LSA | 2h(l + b) | | TSA | 2(lb + bh + hl) | | Space diagonal | √(l² + b² + h²) |
### Cylinder (radius = r, height = h) | Quantity | Formula | |----------|---------| | Volume | πr²h | | CSA | 2πrh | | TSA | 2πr(r + h) |
### Sphere (radius = r) | Quantity | Formula | |----------|---------| | Volume | (4/3)πr³ | | Surface Area | 4πr² |
### Hemisphere (radius = r) | Quantity | Formula | |----------|---------| | Volume | (2/3)πr³ | | CSA | 2πr² | | TSA | 3πr² |
**Unit conversions to remember:**
- 1 m = 100 cm → 1 m² = 10000 cm² → 1 m³ = 1000000 cm³
- 1 litre = 1000 cm³ = 0.001 m³
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Worked Examples
### Example 1 — Cube **Problem:** Find the total surface area and volume of a cube whose edge is 5 cm.
**Solution:**
- TSA = 6a² = 6 × 5² = 6 × 25 = **150 cm²**
- Volume = a³ = 5³ = **125 cm³**
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### Example 2 — Cuboid (Word Problem) **Problem:** A rectangular water tank is 2 m long, 1.5 m wide, and 1 m deep. How many litres of water can it hold?
**Solution:**
- Volume = l × b × h = 2 × 1.5 × 1 = 3 m³
- Convert to litres: 1 m³ = 1000 litres
- Capacity = 3 × 1000 = **3000 litres**
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### Example 3 — Cylinder **Problem:** A cylindrical pillar has radius 14 cm and height 3 m. Find its curved surface area. (Use π = 22/7)
**Solution:**
- First, convert height to cm: 3 m = 300 cm
- CSA = 2πrh = 2 × (22/7) × 14 × 300
- = 2 × 22 × 2 × 300 = **26400 cm²**
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### Example 4 — Sphere **Problem:** Find the volume of a sphere with diameter 21 cm. (Use π = 22/7)
**Solution:**
- Radius = 21/2 = 10.5 cm
- Volume = (4/3)πr³ = (4/3) × (22/7) × (10.5)³
- (10.5)³ = 1157.625
- Volume = (4/3) × (22/7) × 1157.625 = (88/21) × 1157.625 ≈ **4851 cm³**
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Common Mistakes
1. **Confusing surface area with volume**
- Wrong: Using 6a² when asked for volume of a cube.
- Fix: Surface area → square units (cm²); Volume → cubic units (cm³). Read the question carefully.
2. **Mixing up radius and diameter**
- Wrong: Directly using diameter in formulas meant for radius.
- Fix: Always halve the diameter first. If d = 14 cm, then r = 7 cm.
3. **Ignoring unit conversion**
- Wrong: Calculating volume with length in metres and radius in centimetres.
- Fix: Convert all dimensions to the same unit before substituting.
4. **Using wrong formula for LSA vs TSA**
- Wrong: Giving CSA (2πrh) when TSA is asked for a closed cylinder.
- Fix: TSA adds the two circular bases: 2πr² + 2πrh = 2πr(r + h).
5. **Forgetting the fraction in sphere/hemisphere formulas**
- Wrong: Writing volume of sphere as 4πr³ instead of (4/3)πr³.
- Fix: Memorise the fractions: sphere → 4/3; hemisphere → 2/3.
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Quick Reference
- Cube TSA = 6a²; Volume = a³
- Cuboid TSA = 2(lb + bh + hl); Volume = lbh
- Cylinder CSA = 2πrh; TSA = 2πr(r + h); Volume = πr²h
- Sphere SA = 4πr²; Volume = (4/3)πr³
- 1 litre = 1000 cm³; 1 m³ = 1000 litres
- Always check: radius or diameter? Same units throughout?