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Mensuration — Area & Volume

Quantitative Aptitude · Geometry, Mensuration & Trigonometry

Useful for: SSC CGL

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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

The area of a circle is 154 sq cm. Its radius (using pi=22/7) is:
1. 10.5 cm
2. 7 cm
3. 14 cm
4. 3.5 cm
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Answer: (b) 7 cm. Area = pi r^2 = 154 → (22/7)r^2 = 154 → r^2 = 154×7/22 = 49 → r = 7 cm.
The volume of a cube is 216 cubic cm. Its total surface area is:
1. 216 sq cm
2. 196 sq cm
3. 144 sq cm
4. 256 sq cm
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Answer: (a) 216 sq cm. Volume = a^3 = 216 → a = 6 cm.
The diagonal of a square is 10√2 cm. Its area is:
1. 144 sq cm
2. 100 sq cm
3. 200 sq cm
4. 50 sq cm
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Answer: (b) 100 sq cm. Diagonal = a√2 = 10√2 → a = 10.
A cylinder has radius 7 cm and height 10 cm. Its curved surface area (pi=22/7) is:
1. 440 sq cm
2. 220 sq cm
3. 880 sq cm
4. 660 sq cm
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Answer: (a) 440 sq cm. CSA = 2 pi r h = 2×(22/7)×7×10 = 2×22×10 = 440 sq cm.
The volume of a sphere of radius 3 cm (pi=22/7) is approximately:
1. 36 cubic cm
2. 113.14 cubic cm
3. 88 cubic cm
4. 150 cubic cm
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Answer: (b) 113.14 cubic cm. Volume = (4/3) pi r^3 = (4/3)×(22/7)×27 = (4×22×27)/(21) = 2376/21 = 113.14 cubic cm.
The perimeter of a rectangle is 60 cm and its length is 18 cm. Its area is:
1. 216 sq cm
2. 240 sq cm
3. 180 sq cm
4. 200 sq cm
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Answer: (a) 216 sq cm. Perimeter = 2(l+b) = 60 → l+b = 30 → b = 12.
The area of an equilateral triangle with side 6 cm is:
1. 18√3 sq cm
2. 6√3 sq cm
3. 9√3 sq cm
4. 36√3 sq cm
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Answer: (c) 9√3 sq cm. Area = (√3/4)a^2 = (√3/4)×36 = 9√3 sq cm.
A cone has radius 3 cm and height 4 cm. Its slant height is:
1. 5 cm
2. 7 cm
3. 6 cm
4. 4.5 cm
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Answer: (a) 5 cm. Slant l = sqrt(r^2+h^2) = sqrt(9+16) = sqrt(25) = 5 cm.
If the radius of a sphere is doubled, its volume becomes:
1. 4 times
2. 2 times
3. 6 times
4. 8 times
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Answer: (d) 8 times. Volume proportional to r^3.
The total surface area of a hemisphere of radius 7 cm (pi=22/7) is:
1. 462 sq cm
2. 308 sq cm
3. 154 sq cm
4. 616 sq cm
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Answer: (a) 462 sq cm. Hemisphere TSA = 3 pi r^2 = 3×(22/7)×49 = 3×22×7 = 462 sq cm.