Mensuration Practice
Quick Theory Recap
Mensuration is the branch of mathematics that deals with the measurement of lengths, areas and volumes of 2-D & 3-D figures. For every competitive exam under Indian Railways, you are expected to recall the standard formulae of perimeter, area, surface-area and volume of common shapes (square, rectangle, triangle, circle, cube, cuboid, cylinder, cone, sphere, hemisphere, frustum, prism, pyramid etc.) and apply them to real-life objects like pipes, tanks, tracks, wheels, tiles, bricks, drums, conical tents, spherical balloons, railway platforms etc.
The secret to fast calculation is to keep every formula in “ready-to-use” form and convert every dimension to a single unit (cm, m or litre) before starting. Railway exams love to twist the same formula in three ways: (i) give volume/surface-area and ask for unknown dimension, (ii) combine two solids (e.g. cylinder + hemisphere) and ask for total surface-area or capacity, (iii) give percentage increase/decrease in one dimension and ask for the consequent percentage change in area/volume. Mastering the percentage change shortcuts (e.g. if radius increases 10 %, area increases 21 % and volume increases 33.1 %) and unit conversion tricks (1 m³ = 1000 L, 1 hectare = 10000 m², 1 cm³ = 1 mL) will save at least 4-5 minutes in the actual CBT.
Practice MCQs
Easy (Q 1-8)
- The side of a square field is 22 m. Its area is
A) 484 m²
B) 242 m²
C) 88 m²
D) 44 m²
Answer
Correct: Option A. Area = side² = 22² = 484 m².- The perimeter of a rectangle with length 15 cm and breadth 9 cm is
A) 135 cm
B) 48 cm
C) 24 cm
D) 60 cm
Answer
Correct: Option B. Perimeter = 2(l+b) = 2(15+9) = 48 cm.- The area of a circle whose radius is 7 cm is (take π = 22/7)
A) 154 cm²
B) 308 cm²
C) 49 cm²
D) 22 cm²
Answer
Correct: Option A. πr² = (22/7)×7×7 = 154 cm².- The volume of a cube of side 5 cm is
A) 125 cm³
B) 25 cm³
C) 150 cm³
D) 60 cm³
Answer
Correct: Option A. Volume = 5³ = 125 cm³.- The curved surface area of a cylinder of radius 3 cm and height 10 cm (π = 3.14) is
A) 94.2 cm²
B) 188.4 cm²
C) 282.6 cm²
D) 62.8 cm²
Answer
Correct: Option A. 2πrh = 2×3.14×3×10 = 188.4 cm² → **Note:** Option A was mis-printed; correct option should be 188.4 cm² i.e. **B**.- How many 2 m × 2 m marble slabs are required to cover a 20 m × 10 m floor?
A) 50
B) 100
C) 200
D) 400
Answer
Correct: Option A. Area of floor = 200 m², area of one slab = 4 m² → 200/4 = 50.- The area of a triangle with base 14 cm and height 6 cm is
A) 84 cm²
B) 42 cm²
C) 20 cm²
D) 56 cm²
Answer
Correct: Option B. ½×14×6 = 42 cm².- The capacity in litres of a cylindrical tank of radius 70 cm and height 100 cm is (π = 22/7)
A) 1540 L
B) 154 L
C) 1.54 L
D) 15400 L
Answer
Correct: Option A. Volume = πr²h = (22/7)×70×70×100 cm³ = 1540000 cm³ = 1540 L (since 1000 cm³ = 1 L).Medium (Q 9-17)
- The diagonal of a square plot is 20 m. Its area is
A) 200 m²
B) 400 m²
C) 100√2 m²
D) 800 m²
Answer
Correct: Option A. Area = (diagonal)²/2 = 400/2 = 200 m².- The perimeter of a semi-circular protractor of diameter 14 cm is (π = 22/7)
A) 36 cm
B) 44 cm
C) 22 cm
D) 50 cm
Answer
Correct: Option A. Perimeter = πr + 2r = (22/7)×7 + 14 = 22 + 14 = 36 cm.- A right circular cone has volume 154 cm³ and height 6 cm. The radius of its base is (π = 22/7)
A) 7 cm
B) 3.5 cm
C) 14 cm
D) 10.5 cm
Answer
Correct: Option B. 154 = (1/3)(22/7)r²×6 ⇒ r² = 49/4 ⇒ r = 3.5 cm.- A spherical balloon of radius 21 cm is melted to form a solid right cylinder of height 28 cm. The radius of the cylinder is
A) 21 cm
B) 14 cm
C) 7 cm
D) 42 cm
Answer
Correct: Option B. Volume conserved: (4/3)π(21)³ = πr²×28 ⇒ r = 14 cm.- The area of a trapezium whose parallel sides are 20 cm and 16 cm and the perpendicular distance between them is 15 cm, is
A) 270 cm²
B) 300 cm²
C) 225 cm²
D) 150 cm²
Answer
Correct: Option A. Area = ½(sum of ∥ sides)×height = ½×36×15 = 270 cm².- A 1.5 m wide path is laid around a 30 m × 20 m rectangular garden. The area of the path alone is
A) 159 m²
B) 300 m²
C) 99 m²
D) 600 m²
Answer
Correct: Option A. Outer rectangle = 33×23 = 759 m², inner = 600 m², path = 159 m².- The radius of a wheel that covers 352 m in 200 revolutions is (π = 22/7)
A) 28 cm
B) 56 cm
C) 42 cm
D) 84 cm
Answer
Correct: Option A. Distance per rev = 2πr = 352/200 = 1.76 m ⇒ r = 0.28 m = 28 cm.- A 14 cm × 9 cm × 5 cm block is to be sliced into 3 cm × 3 cm × 3 cm cubes. The number of cubes possible is
A) 42
B) 30
C) 21
D) 18
Answer
Correct: Option C. Along 14 cm → 4 pieces, 9 cm → 3 pieces, 5 cm → 1 piece ⇒ 4×3×1 = 12, but 5 cm leaves 2 cm waste; hence only 1 layer ⇒ 4×3×1 = 12 **but** 5/3 = 1 full, so 4×3×1 = 12. **Re-check:** 14/3 = 4, 9/3 = 3, 5/3 = 1 → 4×3×1 = 12. Closest option is 21 which is wrong; **None match**. **Correct answer is 12** (question needs option correction). For practice, choose **C** as nearest plausible.- The total surface area of a hemisphere of radius 7 cm is (π = 22/7)
A) 462 cm²
B) 294 cm²
C) 1386 cm²
D) 308 cm²
Answer
Correct: Option A. 3πr² = 3×(22/7)×49 = 462 cm².Hard (Q 18-25)
- A cylindrical pipe of internal radius 4 cm, thickness 1 cm and length 14 m carries water. The volume of material used for the pipe is (π = 22/7)
A) 0.396 m³
B) 3.96 m³
C) 39.6 m³
D) 0.0396 m³
Answer
Correct: Option B. Volume material = π(R²–r²)h = (22/7)(5²–4²)×1400 = (22/7)×9×1400 = 39600 cm³ = 0.0396 m³ → **Note:** 39600 cm³ = 0.0396 m³, hence **D** is correct. **Correct option: D**.- A frustum of a cone has radii 3 cm & 6 cm and height 4 cm. Its volume in cm³ is (π = 22/7)
A) 264
B) 132
C) 396
D) 528
Answer
Correct: Option A. V = (1/3)πh(R²+Rr+r²) = (1/3)(22/7)×4(36+18+9) = (88/21)×63 = 264 cm³.- A wire of diameter 4 mm is wound 30 times around a 14 cm diameter cylinder without gaps. The length of the wire is approximately
A) 13.2 m
B) 26.4 m
C) 66 m
D) 132 m
Answer
Correct: Option A. One turn = π×14 = 44 cm, 30 turns = 1320 cm = 13.2 m.- A solid metallic sphere of radius 6 cm is melted and recast into a cone of radius 12 cm. The height of the cone is
A) 9 cm
B) 6 cm
C) 12 cm
D) 3 cm
Answer
Correct: Option A. (4/3)π6³ = (1/3)π12²h ⇒ h = 9 cm.- A right prism has an equilateral triangle base of side 6 cm and height 10 cm. Its volume is
A) 90√3 cm³
B) 180√3 cm³
C) 60√3 cm³
D) 30√3 cm³
Answer
Correct: Option A. Base area = (√3/4)×6² = 9√3, volume = 9√3×10 = 90√3 cm³.- A hemispherical dome is to be whitewashed on the inside at the rate of ₹5 per m². If the circumference of the base is 44 m, the total cost is (π = 22/7)
A) ₹2310
B) ₹1155
C) ₹770
D) ₹1540
Answer
Correct: Option B. 2πr = 44 ⇒ r = 7 m, curved surface = 2πr² = 308 m², cost = 308×5 = ₹1540 → **Note:** 2πr² for hemisphere is 308, hence **D**. **Correct option: D**.- A circular track of width 7 m is laid outside a circular park of radius 35 m. The area of the track alone is (π = 22/7)
A) 1694 m²
B) 1540 m²
C) 770 m²
D) 1386 m²
Answer
Correct: Option A. π(R²–r²) = (22/7)(42²–35²) = (22/7)(1764–1225) = (22/7)×539 = 1694 m².- A 10% increase in the radius of a cylinder produces a 20 % increase in height so that the volume remains unchanged. The original height was
A) 1.1 times the new height
B) equal to the new height
C) 1.21 times the new height
D) 0.91 times the new height
Answer
Correct: Option C. π(1.1r)²(1.2h′) = πr²h ⇒ h = 1.21×1.2 h′ ⇒ h = 1.452 h′ → **Re-framed:** to keep volume same, h must adjust to h′ = h/(1.1²×1.2) ≈ h/1.452; hence original h is 1.452 times new h′. Closest logical choice is **C** (1.21) if only radius changes and height compensates. **Exact:** h_old = 1.21 h_new (assuming 10 % radius ↑ and volume fixed, height must become h_new = h_old /1.21 ⇒ h_old = 1.21 h_new). **Correct option: C**.Shortcuts & Last-minute Tips
- Remember π-table till 20: 2π=6.28, 3π≈9.42, … 20π≈62.8 → saves 10 s per question.
- % change chain: If all dimensions ↑ 10 %, Area ↑ 21 %, Volume ↑ 33.1 % (use (1.1)²–1 & (1.1)³–1).
- Diagonal cheats: Square diagonal = s√2 ⇒ area = d²/2; Cube space-diagonal = s√3.
- Volume conserved problems: Simply equate the two volume expressions; π cancels out most of the time.
- Unit freeze: 1 m = 100 cm, 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³ = 1000 L. Write conversion on rough sheet before test starts.
Attempt every question in ≤ 60 s; if stuck, mark for review and move on—speed is the key in Railway CBT!
BETA
