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Mensuration

Mathematical Abilities — Paper-I · Official 2026 Coverage

Useful for: SSC CGL

Overview

Tier-II mensuration combines 2D and 3D shapes, slant heights, recasting and surface-area questions.

Key points

Cone: slant height l = √(r² + h²); curved surface πrl; volume ⅓πr²h.
Cuboid: total surface 2(lb + bh + hl); diagonal √(l² + b² + h²).
Sphere: volume (4/3)πr³; surface 4πr².
Frustum: volume (1/3)πh(R² + r² + Rr).
Recasting: total volume is unchanged.

Explanation

Number of small solids from a big solid = big volume ÷ small volume.
Scaling: areas scale with k², volumes with k³.
Hollow cylinder volume = πh(R² − r²).
Keep π symbolic until the last step when possible.

Step by step

Identify the concept and the formula that links the given values.
Solve step by step, keeping units consistent.
Check the answer against the options and by a quick estimate.

Examples

Cone with r = 3 cm, h = 4 cm: l = 5 cm, curved surface = 15π cm², volume = 12π cm³.
Cuboid 10 × 8 × 6 cm: total surface = 2(80 + 48 + 60) = 376 cm²; diagonal = √200 = 10√2 cm.
A sphere of radius 6 cm is melted into cones of radius 3 cm and height 4 cm: (4/3)π × 216 ÷ (⅓π × 9 × 4) = 288π ÷ 12π = 24 cones.

Traps and advanced points

Trap: Using height for slant height.
Trap: Forgetting that recasting conserves volume only.
Trap: Mixing curved and total surface areas.

Quick revision

l = √(r² + h²).
k² area, k³ volume.
Recast → equal volume.

Practice MCQs

A rectangle is 69 cm × 29 cm. Length rises by 20% and breadth falls by 20%. New area?
1. 1920.96
2. 2305.15
3. 2113.06
4. 1728.86
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Answer: (a) 1920.96. New dimensions = 82.8 and 23.2.
Using π=22/7, find volume of a cylinder with radius 21 cm and height 17 cm.
1. 23562 cm³
2. 25918.2 cm³
3. 28274.4 cm³
4. 21205.8 cm³
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Answer: (a) 23562 cm³. Cylinder volume = πr²h.
Using π=22/7, find volume of a cylinder with radius 70 cm and height 74 cm.
1. 1253560 cm³
2. 1025640 cm³
3. 1139600 cm³
4. 1367520 cm³
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Answer: (c) 1139600 cm³. Cylinder volume = πr²h.
A rectangle is 21 cm × 11 cm. Length rises by 20% and breadth falls by 20%. New area?
1. 199.58
2. 221.76
3. 266.11
4. 243.94
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Answer: (b) 221.76. New dimensions = 25.2 and 8.8.
Using π=22/7, find volume of a cylinder with radius 14 cm and height 6 cm.
1. 4065.6 cm³
2. 3696 cm³
3. 4435.2 cm³
4. 3326.4 cm³
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Answer: (b) 3696 cm³. Cylinder volume = πr²h.
A rectangle is 22 cm × 12 cm. Length rises by 25% and breadth falls by 10%. New area?
1. 267.3
2. 297
3. 356.4
4. 326.7
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Answer: (b) 297. New dimensions = 27.5 and 10.8.
Using π=22/7, find volume of a cylinder with radius 21 cm and height 7 cm.
1. 10672.2 cm³
2. 11642.4 cm³
3. 9702 cm³
4. 8731.8 cm³
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Answer: (c) 9702 cm³. Cylinder volume = πr²h.
A rectangle is 23 cm × 13 cm. Length rises by 10% and breadth falls by 20%. New area?
1. 289.43
2. 236.81
3. 263.12
4. 315.74
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Answer: (c) 263.12. New dimensions = 25.3 and 10.4.
Using π=22/7, find volume of a cylinder with radius 28 cm and height 8 cm.
1. 23654.4 cm³
2. 21683.2 cm³
3. 17740.8 cm³
4. 19712 cm³
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Answer: (d) 19712 cm³. Cylinder volume = πr²h.
A rectangle is 24 cm × 14 cm. Length rises by 20% and breadth falls by 10%. New area?
1. 326.59
2. 399.17
3. 435.46
4. 362.88
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Answer: (d) 362.88. New dimensions = 28.8 and 12.6.
Using π=22/7, find volume of a cylinder with radius 35 cm and height 9 cm.
1. 34650 cm³
2. 31185 cm³
3. 41580 cm³
4. 38115 cm³
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Answer: (a) 34650 cm³. Cylinder volume = πr²h.
A rectangle is 25 cm × 15 cm. Length rises by 25% and breadth falls by 20%. New area?
1. 337.5
2. 412.5
3. 375
4. 450
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Answer: (c) 375. New dimensions = 31.25 and 12.
Using π=22/7, find volume of a cylinder with radius 42 cm and height 10 cm.
1. 60984 cm³
2. 49896 cm³
3. 55440 cm³
4. 66528 cm³
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Answer: (c) 55440 cm³. Cylinder volume = πr²h.
A rectangle is 26 cm × 16 cm. Length rises by 10% and breadth falls by 10%. New area?
1. 370.66
2. 453.02
3. 411.84
4. 494.21
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Answer: (c) 411.84. New dimensions = 28.6 and 14.4.
Using π=22/7, find volume of a cylinder with radius 49 cm and height 11 cm.
1. 91306.6 cm³
2. 83006 cm³
3. 74705.4 cm³
4. 99607.2 cm³
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Answer: (b) 83006 cm³. Cylinder volume = πr²h.