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Mensuration

Quantitative Aptitude / DI · Arithmetic Applications

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Overview

Mensuration is formula recall plus unit care: area in square units, volume in cubic units.

Key points

Circle: area πr², circumference 2πr; rectangle: lb; square: a² or d²/2.
Triangle: ½ × base × height; equilateral triangle: (√3/4)a²; Heron: √[s(s−a)(s−b)(s−c)].
Cylinder: volume πr²h, curved surface 2πrh; cone: ⅓πr²h, curved surface πrl.
Sphere: volume (4/3)πr³, surface 4πr²; hemisphere total surface 3πr².
Cuboid: volume lbh, total surface 2(lb + bh + hl); diagonal √(l² + b² + h²).

Explanation

When a solid is melted and recast, volume is conserved.
If every dimension is multiplied by k, area is multiplied by k² and volume by k³.
Cone slant height l = √(r² + h²).
Choose r as a multiple of 7 to simplify π = 22/7.

Step by step

Name the shape and the quantity asked.
Write the formula.
Substitute in one unit and simplify.

Examples

Cylinder with r = 7 cm, h = 10 cm: volume = 22/7 × 49 × 10 = 1,540 cm³.
A square with diagonal 10 cm has area 10²/2 = 50 cm².
Sphere of radius 21 cm: volume = 4/3 × 22/7 × 9,261 = 38,808 cm³.

Traps and advanced points

Trap: Mixing cm and m in one formula.
Trap: Using total surface area when curved surface is asked (or the reverse).
Trap: Forgetting that recasting conserves volume, not surface area.

Quick revision

Area k², volume k³.
D²/2 for a square.
Melt → equal volumes.

Practice MCQs

Using π=22/7, find volume of a cylinder with radius 7 cm and height 45 cm.
1. 7623 cm³
2. 8316 cm³
3. 6930 cm³
4. 6237 cm³
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Answer: (c) 6930 cm³. Cylinder volume = πr²h.
Using π=22/7, find volume of a cylinder with radius 70 cm and height 54 cm.
1. 748440 cm³
2. 831600 cm³
3. 914760 cm³
4. 997920 cm³
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Answer: (b) 831600 cm³. Cylinder volume = πr²h.
A rectangle is 106 cm × 36 cm. Length rises by 25% and breadth falls by 10%. New area?
1. 5151.6
2. 3863.7
3. 4722.3
4. 4293
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Answer: (d) 4293. New dimensions = 132.5 and 32.4.
A rectangle is 117 cm × 17 cm. Length rises by 20% and breadth falls by 20%. New area?
1. 1718.5
2. 2100.38
3. 2291.33
4. 1909.44
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Answer: (d) 1909.44. New dimensions = 140.4 and 13.6.
A rectangle is 28 cm × 18 cm. Length rises by 25% and breadth falls by 10%. New area?
1. 510.3
2. 567
3. 623.7
4. 680.4
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Answer: (b) 567. New dimensions = 35 and 16.2.
A rectangle is 56 cm × 16 cm. Length rises by 10% and breadth falls by 10%. New area?
1. 1064.45
2. 798.34
3. 975.74
4. 887.04
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Answer: (d) 887.04. New dimensions = 61.6 and 14.4.
A rectangle is 108 cm × 38 cm. Length rises by 20% and breadth falls by 10%. New area?
1. 4875.55
2. 4432.32
3. 3989.09
4. 5318.78
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Answer: (b) 4432.32. New dimensions = 129.6 and 34.2.
A rectangle is 86 cm × 16 cm. Length rises by 10% and breadth falls by 10%. New area?
1. 1634.69
2. 1498.46
3. 1362.24
4. 1226.02
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Answer: (c) 1362.24. New dimensions = 94.6 and 14.4.
A rectangle is 32 cm × 22 cm. Length rises by 10% and breadth falls by 10%. New area?
1. 627.26
2. 696.96
3. 766.66
4. 836.35
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Answer: (b) 696.96. New dimensions = 35.2 and 19.8.
A rectangle is 75 cm × 35 cm. Length rises by 20% and breadth falls by 20%. New area?
1. 2772
2. 3024
3. 2268
4. 2520
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Answer: (d) 2520. New dimensions = 90 and 28.
Using π=22/7, find volume of a cylinder with radius 28 cm and height 18 cm.
1. 39916.8 cm³
2. 44352 cm³
3. 53222.4 cm³
4. 48787.2 cm³
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Answer: (b) 44352 cm³. Cylinder volume = πr²h.
A rectangle is 102 cm × 32 cm. Length rises by 20% and breadth falls by 10%. New area?
1. 4230.14
2. 3172.61
3. 3525.12
4. 3877.63
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Answer: (c) 3525.12. New dimensions = 122.4 and 28.8.
A rectangle is 61 cm × 21 cm. Length rises by 25% and breadth falls by 20%. New area?
1. 1152.9
2. 1537.2
3. 1409.1
4. 1281
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Answer: (d) 1281. New dimensions = 76.25 and 16.8.
A rectangle is 55 cm × 15 cm. Length rises by 25% and breadth falls by 20%. New area?
1. 825
2. 990
3. 742.5
4. 907.5
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Answer: (a) 825. New dimensions = 68.75 and 12.
Using π=22/7, find volume of a cylinder with radius 56 cm and height 52 cm.
1. 461260.8 cm³
2. 512512 cm³
3. 563763.2 cm³
4. 615014.4 cm³
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Answer: (b) 512512 cm³. Cylinder volume = πr²h.