Mensuration
SSC-CGL Exams
1. Introduction
Mensuration deals with length, area, surface area, and volume. SSC CGL questions are mostly formula-based, but visualization helps solve faster and avoid errors.
We divide it into:
1. 2D Geometry
Squares, rectangles, triangles, circles, polygons
2. 3D Geometry
Cubes, cuboids, cylinders, cones, spheres
2. 2D Geometry (Plane Figures)
Squares & Rectangles
| Shape | Area | Perimeter | Notes |
|---|---|---|---|
| Square | A = a² | P = 4a | a = side |
| Rectangle | A = l × b | P = 2(l+b) | l = length, b = breadth |
Example 1
Square side = 8 cm → Area?
A = 8² = 64 cm²
64 cm²
Triangle
| Type | Area Formula | Perimeter / Notes |
|---|---|---|
| Any triangle | A = ½ × base × height | - |
| Equilateral | A = (√3/4)a² | Perimeter = 3a |
| Right-angled | A = ½ × perpendicular × base | Hypotenuse = c |
Example 2
Right triangle, base = 6 cm, height = 8 cm → Area
A = ½ × 6 × 8 = 24 cm²
Circle
| Property | Formula |
|---|---|
| Circumference | C = 2πr |
| Area | A = πr² |
| Diameter | d = 2r |
Example 3
Radius = 7 cm → Circumference & Area
C = 2π7 = 14π, A = π7² = 49π
3. 3D Geometry (Solids)
Cube
Volume: V = a³
Surface Area: SA = 6a²
Cuboid (Rectangular Box)
Volume: V = l × b × h
Surface Area: SA = 2(lb + bh + hl)
Cylinder
Volume: V = πr²h
Curved Surface Area: CSA = 2πrh
Total Surface Area: TSA = 2πr(h+r)
Cone
Volume: V = ⅓πr²h
Curved Surface Area: CSA = πrl
(l = slant height)
Total Surface Area: TSA = πr(l + r)
Sphere / Hemisphere
| Shape | Volume | Surface Area |
|---|---|---|
| Sphere | V = 4/3 πr³ | SA = 4πr² |
| Hemisphere | V = 2/3 πr³ | SA = 3πr² |
4. Visualization Tips
Draw a quick sketch — even rough — for each SSC problem.
Label dimensions (r, h, l, b, a) before substituting formulas.
Remember CSA vs TSA: CSA = only curved part; TSA = CSA + base(s).
For combined solids (cylinder + hemisphere), calculate separately, then sum.
5. Detailed Examples
Example 4: Cube & Cuboid
Cube side = 5 cm
V = 5³ = 125 cm³
SA = 6 × 25 = 150 cm²
Cuboid l=6, b=4, h=3
V = 6×4×3 = 72 cm³
SA = 2(24 + 12 + 18) = 108 cm²
Example 5: Cylinder
Radius = 7 cm, height = 10 cm
V = π7² × 10 = 490π cm³
CSA = 2π7×10 = 140π cm²
TSA = 2π7(10+7) = 238π cm²
Example 6: Cone
r = 5 cm, h = 12 cm
l = √(r² + h²) = √(25 + 144) = 13
CSA = πrl = 5π×13 = 65π cm²
TSA = πr(l + r) = π5(13+5) = 90π cm²
Example 7: Sphere / Hemisphere
r = 7 cm
Sphere: V = 4/3π343 ≈ 478.67π, SA = 4π49 = 196π
Hemisphere: V = 2/3π343 ≈ 228.67π, SA = 3π49 = 147π
6. SSC Short Tricks / Tips
Always check units: cm, m, or mm. Convert first.
For CSA vs TSA: visualize curved part only vs curved + base(s).
Combined solids → add volumes / surfaces.
Circle → circumference ~ 3.14 × 2r; area ~ 3.14 × r²
Memorize formula triangles (cube, cuboid, cylinder) for quick recall.
7. Practice Section
Q1. Square side = 12 cm → Area & Perimeter
View Answer
A = 12² = 144 cm², P = 4×12 = 48 cm
144 cm², 48 cm
Q2. Triangle base = 10 cm, height = 6 cm → Area
View Answer
A = ½ × 10 × 6 = 30 cm²
30 cm²
Q3. Circle radius = 14 cm → Area & Circumference
View Answer
A = π14² = 196π, C = 2π14 = 28π
196π, 28π
Q4. Cylinder r = 7 cm, h = 15 cm → Volume & CSA & TSA
View Answer
V = π7²×15 = 735π, CSA = 2π7×15=210π, TSA=2π7(15+7)=308π
735π, 210π, 308π
Q5. Cone r = 3 cm, h = 4 cm → l, CSA, TSA
View Answer
l = √(3²+4²)=5, CSA = π×3×5=15π, TSA=π3(5+3)=24π
l=5, CSA=15π, TSA=24π
Q6. Sphere radius = 10 cm → Volume & SA
View Answer
V = 4/3 π1000 ≈ 4186.67 cm³, SA = 4π100=400π
4186.67 cm³, 400π cm²
Q7. Hemisphere radius = 7 cm → Volume & SA
View Answer
V = 2/3 π343 ≈ 228.67π, SA = 3π49 = 147π
228.67π, 147π
8. Quick Recap Table
| Shape | Area / Volume | Notes |
|---|---|---|
| Square | a² | Perimeter = 4a |
| Rectangle | l×b | P = 2(l+b) |
| Triangle | ½ bh | Equilateral = √3/4 a² |
| Circle | πr², 2πr | - |
| Cube | a³, 6a² | V, SA |
| Cuboid | lbh, 2(lb+bh+hl) | V, SA |
| Cylinder | πr²h, 2πrh, 2πr(h+r) | V, CSA, TSA |
| Cone | ⅓πr²h, πrl, πr(l+r) | l = √(r²+h²) |
| Sphere | 4/3πr³, 4πr² | - |
| Hemisphere | 2/3πr³, 3πr² | - |
You've completed Article 10: Mensuration (2D & 3D Geometry)!
Courage Tip: Draw the solid or figure whenever possible. SSC loves combined solids, pipes, and circles inside squares, where visualization is key to faster solutions.
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