Area & Volume
Overview
Area and Volume is a scoring topic in Maharashtra Police Bharti Mathematics. Questions typically ask you to calculate the surface area or volume of basic 3-D shapes — cube, cuboid, sphere, cylinder, and occasionally cone or hemisphere. Mastering 8–10 formulas and understanding when to apply each is the key to solving these problems quickly.
This topic connects directly to real-life applications: painting walls (surface area), filling tanks (volume), and material estimation. Students who memorise formulas but skip unit conversion practice lose easy marks here.
Focus on: (1) memorising formulas with correct notation, (2) identifying which formula applies to a given scenario, and (3) careful unit handling — cm to m, litres to cubic cm, etc.
Key Concepts
- Surface Area measures the total outer covering of a 3-D object — think of how much paint is needed. Measured in square units (cm², m²).
- Volume measures the space inside a 3-D object — think of how much water it can hold. Measured in cubic units (cm³, m³).
- Lateral/Curved Surface Area (LSA/CSA) excludes the top and bottom faces. Use this when a container is open or when only the sides are painted.
- Total Surface Area (TSA) includes all faces — top, bottom, and sides.
- 1 litre = 1000 cm³ = 0.001 m³ — This conversion appears in almost every tank/container problem.
- When dimensions are in different units, convert all to the same unit first before applying any formula.
- For hollow objects (pipes, shells), subtract inner volume/area from outer volume/area.
Formulas / Key Facts
Cube (all sides equal, side = a)
| Measure | Formula |
|---|---|
| Volume | a³ |
| LSA (4 faces) | 4a² |
| TSA (6 faces) | 6a² |
| Diagonal | a√3 |
Cuboid (length = l, breadth = b, height = h)
| Measure | Formula |
|---|---|
| Volume | l × b × h |
| LSA (4 walls) | 2h(l + b) |
| TSA (all 6 faces) | 2(lb + bh + hl) |
| Diagonal | √(l² + b² + h²) |
Cylinder (radius = r, height = h)
| Measure | Formula |
|---|---|
| Volume | πr²h |
| CSA (curved only) | 2πrh |
| TSA (curved + 2 circles) | 2πr(r + h) |
Sphere (radius = r)
| Measure | Formula |
|---|---|
| Volume | (4/3)πr³ |
| Surface Area | 4πr² |
Hemisphere (radius = r)
| Measure | Formula |
|---|---|
| Volume | (2/3)πr³ |
| CSA (curved only) | 2πr² |
| TSA (curved + flat circle) | 3πr² |
Cone (radius = r, height = h, slant height = l)
| Measure | Formula |
|---|---|
| Volume | (1/3)πr²h |
| Slant height | l = √(r² + h²) |
| CSA | πrl |
| TSA | πr(r + l) |
Use π = 22/7 or 3.14 as given in the question. If not specified, 22/7 is standard for Police Bharti.
Worked Examples
Example 1: Cube Problem
Q: Find the volume and TSA of a cube with side 7 cm.
Solution:
- Volume = a³ = 7³ = 343 cm³
- TSA = 6a² = 6 × 7² = 6 × 49 = 294 cm²
Answer: Volume = 343 cm³, TSA = 294 cm²
Example 2: Cylinder with Unit Conversion
Q: A cylindrical tank has radius 1.4 m and height 2 m. How many litres of water can it hold? (π = 22/7)
Solution:
- Volume = πr²h = (22/7) × (1.4)² × 2
- = (22/7) × 1.96 × 2
- = (22/7) × 3.92
- = 22 × 0.56 × 2 = 12.32 m³
Convert to litres: 1 m³ = 1000 litres
- Volume = 12.32 × 1000 = 12320 litres
Answer: 12320 litres
Example 3: Sphere Melted into Smaller Spheres
Q: A solid sphere of radius 6 cm is melted and recast into smaller spheres of radius 2 cm. How many smaller spheres are formed?
Solution:
- Volume of large sphere = (4/3)πR³ = (4/3)π(6)³ = (4/3)π × 216 = 288π cm³
- Volume of small sphere = (4/3)πr³ = (4/3)π(2)³ = (4/3)π × 8 = (32/3)π cm³
- Number of small spheres = 288π ÷ (32/3)π = 288 × (3/32) = 864/32 = 27
Answer: 27 small spheres
Example 4: Painting the Walls (LSA Application)
Q: A room is 10 m long, 8 m broad, and 4 m high. Find the cost of painting its four walls at Rs 15 per m².
Solution:
- LSA of cuboid (4 walls) = 2h(l + b) = 2 × 4 × (10 + 8) = 8 × 18 = 144 m²
- Cost = 144 × 15 = Rs 2160
Answer: Rs 2160
Common Mistakes
- Confusing LSA with TSA → LSA excludes top/bottom; TSA includes all faces. Read the question: "walls only" = LSA, "entire surface" = TSA.
- Forgetting to square or cube the radius → In πr²h, the radius is squared. In (4/3)πr³, it is cubed. Write formulas carefully.
- Mixing radius and diameter → Questions often give diameter. Always halve it to get radius before applying formulas.
- Ignoring unit conversion → If radius is in cm and height in m, convert both to same unit first. Final answer unit must match what is asked (litres vs cm³).
- Using wrong value of π → Check if question specifies 3.14 or 22/7. Using the wrong value gives wrong numerical answer.
- Hollow cylinder volume → For pipes, Volume = π(R² − r²)h where R = outer radius, r = inner radius. Don't just use one radius.
Quick Reference
- Cube: V = a³, TSA = 6a², Diagonal = a√3
- Cuboid: V = lbh, TSA = 2(lb + bh + hl), LSA = 2h(l + b)
- Cylinder: V = πr²h, CSA = 2πrh, TSA = 2πr(r + h)
- Sphere: V = (4/3)πr³, SA = 4πr²
- 1 m³ = 1000 litres = 10⁶ cm³
- When shape is melted and recast: Total volume remains same