Mensuration Formulas for SSC CGL (Quick Answer)
The mensuration formulas asked most in SSC CGL are: circle area πr², triangle area ½ × base × height (or Heron's √[s(s−a)(s−b)(s−c)]), cylinder volume πr²h, cone volume ⅓πr²h, sphere volume (4/3)πr³ and sphere surface area 4πr². SSC usually takes π = 22/7, so pick radii like 7, 14 or 21 and most answers come out as whole numbers.
Key takeaway: Learn the 2D table (area and perimeter) and the 3D table (volume, curved and total surface area) below, plus three shortcuts: equilateral triangle area = (√3/4)a², cube diagonal = a√3, and "if every length grows by x%, area grows by (2x + x²/100)% and volume by the cube of the growth factor". These cover most Tier 1 mensuration questions.
SSC CGL Tier 1 normally has one to three mensuration questions and Tier 2 several more, often mixed with geometry. Every solved value in this note was computed in Python with π = 22/7.
2D Mensuration Formulas for SSC CGL (Area and Perimeter)
| Shape | Area | Perimeter / other |
|---|---|---|
| Square (side a) | a² | 4a; diagonal a√2 |
| Rectangle (l, b) | l × b | 2(l + b); diagonal √(l² + b²) |
| Triangle | ½ × base × height | a + b + c |
| Triangle (Heron) | √[s(s − a)(s − b)(s − c)], s = (a + b + c)/2 | — |
| Equilateral triangle (side a) | (√3/4)a² | 3a; height (√3/2)a |
| Right triangle (legs b, h) | ½ × b × h | hypotenuse √(b² + h²) |
| Parallelogram | base × height | 2(a + b) |
| Rhombus (diagonals d₁, d₂) | ½ × d₁ × d₂ | side = ½√(d₁² + d₂²) |
| Trapezium (parallel sides a, b; height h) | ½ × (a + b) × h | sum of sides |
| Circle (radius r) | πr² | 2πr |
| Semicircle | ½πr² | πr + 2r |
| Sector (angle θ) | (θ/360) × πr² | arc = (θ/360) × 2πr |
| Ring (radii R, r) | π(R² − r²) | — |
| Regular hexagon (side a) | (3√3/2)a² | 6a |
A common error in some formula sheets is writing the rectangle diagonal as l² × b². The correct relation is diagonal² = l² + b².
3D Mensuration Formulas for SSC CGL (Volume and Surface Area)
| Solid | Volume | Curved / lateral surface area | Total surface area |
|---|---|---|---|
| Cube (side a) | a³ | 4a² | 6a²; diagonal a√3 |
| Cuboid (l, b, h) | lbh | 2h(l + b) | 2(lb + bh + hl); diagonal √(l² + b² + h²) |
| Cylinder (r, h) | πr²h | 2πrh | 2πr(r + h) |
| Hollow cylinder (R, r, h) | πh(R² − r²) | 2πh(R + r) | 2πh(R + r) + 2π(R² − r²) |
| Cone (r, h, slant l) | ⅓πr²h | πrl, where l = √(r² + h²) | πr(l + r) |
| Frustum (radii R, r, height h) | ⅓πh(R² + r² + Rr) | π(R + r)l | π(R + r)l + πR² + πr² |
| Sphere (r) | (4/3)πr³ | — | 4πr² |
| Hemisphere (r) | (2/3)πr³ | 2πr² | 3πr² |
| Prism | base area × height | base perimeter × height | lateral + 2 × base |
| Pyramid | ⅓ × base area × height | ½ × base perimeter × slant height | lateral + base |
Mensuration Shortcuts for SSC CGL (Percentage Change)
| Change | Effect |
|---|---|
| Every side of a 2D figure +x% | Area changes by (2x + x²/100)% |
| Radius of a circle +20% | Area +44% |
| Every edge of a cube +10% | Volume × 1.1³ = +33.1% |
| Radius ratio of spheres r₁ : r₂ | Volume ratio r₁³ : r₂³, surface ratio r₁² : r₂² |
| Solid melted and recast | Volume stays the same |
| Cone and cylinder with same r, h | Cone volume = ⅓ of cylinder |
| Sphere inside a cylinder (same r, h = 2r) | Sphere volume = ⅔ of cylinder |
Mensuration Questions for SSC CGL with Solutions (2D)
Q1. Find the area of an equilateral triangle of side 12 cm. (√3/4) × 144 = 36√3 cm² (about 62.35 cm²).
Q2. The sides of a triangle are 13, 14 and 15 cm. Find its area. s = 21. Area = √(21 × 8 × 7 × 6) = √7056 = 84 cm².
Q3. The circumference of a circle is 88 cm. Find its area. 2 × 22/7 × r = 88, so r = 14. Area = 22/7 × 196 = 616 cm².
Q4. Find the area and arc length of a sector of radius 21 cm and angle 60°. Area = (60/360) × 22/7 × 441 = 231 cm². Arc = (60/360) × 2 × 22/7 × 21 = 22 cm.
Q5. The diagonals of a rhombus are 16 cm and 12 cm. Find its area and side. Area = ½ × 16 × 12 = 96 cm². Side = ½√(256 + 144) = 10 cm.
Q6. A trapezium has parallel sides 10 m and 16 m and height 8 m. Find its area. ½ × 26 × 8 = 104 m².
Q7. A circular path 7 m wide runs around a circular lawn of radius 35 m. Find the area of the path. π(42² − 35²) = 22/7 × 539 = 1694 m².
Mensuration Questions for SSC CGL with Solutions (3D)
Q8. A cuboid is 12 cm × 4 cm × 3 cm. Find its volume, total surface area and diagonal. Volume = 144 cm³. TSA = 2(48 + 12 + 36) = 192 cm². Diagonal = √(144 + 16 + 9) = 13 cm.
Q9. A cylinder has radius 7 cm and height 10 cm. Find its volume, CSA and TSA. Volume = 22/7 × 49 × 10 = 1540 cm³. CSA = 2 × 22/7 × 7 × 10 = 440 cm². TSA = 2 × 22/7 × 7 × 17 = 748 cm².
Q10. A cone has radius 7 cm and height 24 cm. Find its slant height, volume and CSA. l = √(49 + 576) = 25 cm. Volume = ⅓ × 22/7 × 49 × 24 = 1232 cm³. CSA = 22/7 × 7 × 25 = 550 cm².
Q11. Find the volume and surface area of a sphere of radius 21 cm. Volume = 4/3 × 22/7 × 9261 = 38,808 cm³. Surface area = 4 × 22/7 × 441 = 5544 cm².
Q12. Find the total surface area of a solid hemisphere of radius 7 cm. 3 × 22/7 × 49 = 462 cm².
Q13. A sphere of radius 6 cm is melted and recast into a cone of radius 6 cm. Find the height of the cone. (4/3)π × 216 = ⅓π × 36 × h, so h = 24 cm.
Q14. Find the volume of a frustum with radii 14 cm and 7 cm and height 6 cm. ⅓ × 22/7 × 6 × (196 + 49 + 98) = 2156 cm³.
Q15. Each edge of a cube is increased by 10%. By what percent does its volume increase? 1.1³ = 1.331, so 33.1%.
If you like these, the circle and triangle facts behind them are in our trigonometry formulas for SSC CGL, and the speed tricks are in the Vedic maths and quant shortcuts note.
Common Mistakes in Mensuration Questions
- Using TSA when the question asks for CSA (or the reverse), especially for cylinders and hemispheres.
- Forgetting the slant height in cone CSA: it is πrl, not πrh.
- Mixing units: convert cm and m before you calculate (1 m³ = 1000 litres).
- Treating a hollow hemisphere bowl's area as 3πr²; an open bowl's outer curved area is 2πr².
- Writing the rectangle diagonal as l² × b² instead of √(l² + b²).
Units Conversion Table for Mensuration
| Convert | Value |
|---|---|
| 1 m | 100 cm |
| 1 m² | 10,000 cm² |
| 1 m³ | 10,00,000 cm³ = 1000 litres |
| 1 litre | 1000 cm³ |
| 1 hectare | 10,000 m² |
| 1 km² | 100 hectares |
Mensuration Quick Revision Table
| If the question says | Use |
|---|---|
| "Paint the walls of a room" | Lateral area of a cuboid, 2h(l + b), minus doors and windows |
| "Cost of fencing" | Perimeter × rate |
| "Water in a tank" | Volume, then convert to litres |
| "Melted and recast" | Equate volumes |
| "Open box" or "bowl" | Leave out the top face |
| "Wire bent into a shape" | Perimeters are equal |
| "Path around a lawn" | Ring area π(R² − r²) or outer rectangle − inner rectangle |
Mensuration Practice Questions for SSC CGL (with Answers)
- The diagonal of a square is 10√2 cm. Area? Answer: 100 cm².
- Perimeter of a semicircle of radius 14 cm? Answer: 72 cm (44 + 28).
- A cube has volume 343 cm³. Total surface area? Answer: 294 cm² (side 7).
- A cylinder of radius 7 cm has volume 1540 cm³. Height? Answer: 10 cm.
- CSA of a cone with radius 3 cm and height 4 cm? Answer: 15π cm² (l = 5).
- Radii of two spheres are in the ratio 2 : 3. Ratio of volumes? Answer: 8 : 27.
- An equilateral triangle has perimeter 36 cm. Area? Answer: 36√3 cm².
- Diagonal of a 24 cm × 7 cm rectangle? Answer: 25 cm.
- Volume of a hemispherical bowl of radius 21 cm? Answer: 19,404 cm³.
- The radius of a circle is increased by 20%. Increase in area? Answer: 44%.
Test yourself in the SSC Geometry and Mensuration Mock Test and the SSC CGL Tier 1 Geometry Practice Set.
Mensuration Formulas FAQs
What value of π should I use in SSC CGL?
Use 22/7 unless the question says otherwise. SSC chooses radii like 7, 14, 21 and 35 so that 22/7 cancels neatly.
How many mensuration questions come in SSC CGL Tier 1?
Usually one to three, with more in Tier 2. Mensuration is often combined with geometry or percentage change.
What is the difference between CSA and TSA?
CSA (curved or lateral surface area) leaves out the flat faces. TSA adds them: for a cylinder, TSA = CSA + 2πr².
What happens to volume when a solid is melted and recast?
The volume stays the same, so equate the two volume formulas and solve.
Which 3D formulas are most important for SSC CGL?
Cylinder, cone, sphere and hemisphere volume and surface area, plus cube and cuboid diagonals.
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