Area and Volume
Overview
Area and Volume (Mensuration) is a high-scoring topic in TNPSC Group IV Aptitude section. Questions test your ability to calculate surface measurements of flat shapes and space occupied by solid objects. This topic directly applies SSLC-level geometry formulas, making it accessible if you memorize the key formulas and practice unit conversions.
The exam often combines shapes—like finding the area of a path around a garden or volume of water in a cylindrical tank. Mastering this topic requires formula recall, careful unit handling, and recognizing which formula applies to each shape.
Key Concepts
- Area measures the surface covered by a 2D shape, expressed in square units (cm², m², km²).
- Perimeter is the total boundary length of a 2D figure, expressed in linear units (cm, m).
- Volume measures the space occupied by a 3D object, expressed in cubic units (cm³, m³, litres).
- Surface Area of 3D objects has two types: Curved Surface Area (CSA) covers only the curved part; Total Surface Area (TSA) includes all faces.
- π (pi) is taken as 22/7 or 3.14 unless specified otherwise in the question.
- Unit conversion is critical: 1 m = 100 cm, 1 m² = 10000 cm², 1 m³ = 1000 litres, 1 litre = 1000 cm³.
- For composite figures, break the shape into standard parts, calculate separately, then add or subtract as needed.
- Path/Border problems: Area of path = Area of outer figure − Area of inner figure.
Formulas / Key Facts
2D Figures
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | l × b | 2(l + b) |
| Square | a² | 4a |
| Triangle | ½ × base × height | Sum of three sides |
| Equilateral Triangle | (√3/4) × a² | 3a |
| Circle | πr² | 2πr (Circumference) |
| Semicircle | ½πr² | πr + 2r |
| Parallelogram | base × height | 2(a + b) |
| Trapezium | ½ × (sum of parallel sides) × height | Sum of all sides |
| Rhombus | ½ × d₁ × d₂ | 4a |
3D Figures
| Shape | Volume | CSA | TSA |
|---|---|---|---|
| Cube | a³ | 4a² | 6a² |
| Cuboid | l × b × h | 2h(l + b) | 2(lb + bh + hl) |
| Cylinder | πr²h | 2πrh | 2πr(r + h) |
| Cone | ⅓πr²h | πrl (l = slant height) | πr(r + l) |
| Sphere | (4/3)πr³ | 4πr² | 4πr² |
| Hemisphere | (2/3)πr³ | 2πr² | 3πr² |
Slant height of cone: l = √(r² + h²)
Diagonal of cuboid: √(l² + b² + h²)
Diagonal of cube: a√3
Worked Examples
Example 1: Area of a Rectangular Path
A rectangular garden is 40 m long and 30 m wide. A path 2 m wide runs outside around it. Find the area of the path.
Solution:
- Outer dimensions: Length = 40 + 2 + 2 = 44 m, Breadth = 30 + 2 + 2 = 34 m
- Area of outer rectangle = 44 × 34 = 1496 m²
- Area of inner garden = 40 × 30 = 1200 m²
- Area of path = 1496 − 1200 = 296 m²
Example 2: Volume of a Cylinder
A cylindrical tank has radius 7 m and height 10 m. Find its capacity in litres.
Solution:
- Volume = πr²h = (22/7) × 7 × 7 × 10
- Volume = 22 × 7 × 10 = 1540 m³
- Converting: 1 m³ = 1000 litres
- Capacity = 1540 × 1000 = 15,40,000 litres
Example 3: Total Surface Area of a Cone
A cone has radius 3 cm and height 4 cm. Find its total surface area.
Solution:
- First find slant height: l = √(r² + h²) = √(9 + 16) = √25 = 5 cm
- TSA = πr(r + l) = (22/7) × 3 × (3 + 5)
- TSA = (22/7) × 3 × 8 = (22 × 24)/7 = 528/7 = 75.43 cm²
Example 4: Combined Shapes
A solid is made of a hemisphere mounted on a cylinder. Both have radius 7 cm. If the cylinder height is 10 cm, find the total volume.
Solution:
- Volume of cylinder = πr²h = (22/7) × 49 × 10 = 1540 cm³
- Volume of hemisphere = (2/3)πr³ = (2/3) × (22/7) × 343 = (2 × 22 × 49)/3 = 2156/3 = 718.67 cm³
- Total volume = 1540 + 718.67 = 2258.67 cm³
Common Mistakes
- Confusing radius with diameter → Always check: if diameter is given, divide by 2 to get radius before applying formulas.
- Forgetting to square or cube units during conversion → 1 m² = 10000 cm² (not 100), 1 m³ = 1000000 cm³ (not 1000). Convert dimensions first, then calculate.
- Using wrong formula for CSA vs TSA → CSA excludes flat faces (top/bottom), TSA includes everything. Read the question carefully for "curved" vs "total."
- Mixing up cone and cylinder formulas → Cone volume has the factor ⅓; cylinder does not. Cone CSA uses slant height (l), not vertical height (h).
- Ignoring path direction (inside vs outside) → For paths outside, add width twice to each dimension. For paths inside, subtract twice.
- Not calculating slant height for cones → If only r and h are given, you must calculate l = √(r² + h²) before finding CSA or TSA.
Quick Reference
- Rectangle area = l × b; Perimeter = 2(l + b)
- Circle area = πr²; Circumference = 2πr
- Cylinder volume = πr²h; Sphere volume = (4/3)πr³
- Cone volume = ⅓πr²h; Slant height l = √(r² + h²)
- Path area = Outer area − Inner area
- 1 m³ = 1000 litres; always convert units before calculating