Mensuration for IBPS Clerk

Mensuration is not a big block in the clerk exam, but when a question comes it is usually a quick one if you know the formula and slow if you have to work it out. That makes it a good topic to learn once, properly, and then keep warm with a few questions a week.

Clerk-level questions are mostly direct: find the area of a field, the cost of fencing, the volume of a tank. The twists are predictable. A path runs around a lawn. The sides of a figure grow by some percentage. A solid is melted and recast into another shape. Each twist has a standard way in, covered below.

Why the formulas look the way they do

You do not have to derive formulas in the exam, but knowing where they come from helps you remember them and spot errors.

• Area counts unit squares. A 12 m × 8 m rectangle holds 12 rows of 8 one-metre squares, so its area is 96 m².
• A triangle is half a rectangle (or half a parallelogram) with the same base and height. Hence ½ × base × height.
• Volume of any prism or cylinder = base area × height. Stack identical slices. A cylinder is a stack of circles, so πr² × h.
• A cone holds one-third of the cylinder with the same base and height. Hence ⅓πr²h.
• Scaling: if every length is multiplied by k, area is multiplied by k² and volume by k³. Double the radius of a sphere and its volume becomes eight times as large.

2D shapes

Use π = 22/7 when the radius is a multiple of 7, and 3.14 otherwise, unless the options are written in terms of π.

3D shapes

Units to keep straight: 1 m² = 10,000 cm². 1 m³ = 1,000 litres. 1 litre = 1,000 cm³. 1 hectare = 10,000 m².

The standard twists

• A path outside a rectangle of width w: the outer rectangle is (l + 2w) by (b + 2w). Path area = outer area − inner area.
• Percentage change in area: if one side changes by x% and the other by y%, area changes by x + y + xy/100 percent. For a square both are the same x.
• Melting and recasting: the volume stays the same. Set old volume = new volume (× the number of new pieces).

Worked examples

Example 1. A circle has radius 7 cm. Find its area and circumference.

1. Area = 22/7 × 7 × 7 = 154 cm².
2. Circumference = 2 × 22/7 × 7 = 44 cm.

Example 2. A lawn is 20 m by 15 m. A path 2.5 m wide runs around it on the outside. Find the area of the path and the cost of paving it at ₹20 per m².

1. Outer rectangle = (20 + 5) × (15 + 5) = 25 × 20 = 500 m².
2. Path area = 500 − (20 × 15) = 500 − 300 = 200 m².
3. Cost = 200 × 20 = ₹4,000.

Example 3. Each side of a square is increased by 20%. By what percentage does its area increase? What if a rectangle's length rises by 20% and its breadth falls by 10%?

1. Square: 20 + 20 + (20 × 20)/100 = 44%. Check: 1.2 × 1.2 = 1.44.
2. Rectangle: 20 − 10 + (20 × −10)/100 = 20 − 10 − 2 = 8% increase. Check: 1.2 × 0.9 = 1.08.

Example 4. A cuboid is 10 cm × 6 cm × 4 cm. Find its volume and total surface area.

1. Volume = 10 × 6 × 4 = 240 cm³.
2. Surface area = 2(60 + 24 + 40) = 2 × 124 = 248 cm².

Example 5. A cylindrical tank has radius 7 cm and height 10 cm. Find its volume in litres and its curved surface area.

1. Volume = 22/7 × 49 × 10 = 1,540 cm³ = 1.54 litres.
2. Curved surface = 2 × 22/7 × 7 × 10 = 440 cm².

Example 6. A metal sphere of radius 6 cm is melted and recast into a cylinder of radius 4 cm. Find the height of the cylinder.

1. Sphere volume = (4/3) × π × 216 = 288π.
2. Cylinder volume = π × 16 × h.
3. 16h = 288, so h = 18 cm. Keeping π as a symbol saved all the multiplication.

Common mistakes

• Using the diameter where the formula needs the radius.
• Using the height of a cone instead of its slant height for the curved surface.
• Mixing units, for example a tank in metres and an answer asked in litres.
• Adding percentage changes in area (20% + 20% = 40%) instead of multiplying the factors.

Practice

Set a six-minute timer.

1. A rectangle is 15 m by 10 m. Find its area and perimeter.
2. A circle's circumference is 88 cm. Find its area.
3. A cube has edge 5 cm. Find its volume and total surface area.
4. A right-angled triangle has base 6 cm and height 8 cm. Find its area and hypotenuse.
5. A cone has radius 3 cm and height 4 cm. Find its slant height, curved surface area and volume in terms of π.
6. The radius of a circle is increased by 10%. By what percentage does its area increase?
7. A tank is 2 m long, 1.5 m wide and 1 m deep. How many litres does it hold?
8. The diagonal of a square is 12 cm. Find its area.

Answers:

1. 150 m² and 50 m. 15 × 10; 2 × 25.
2. 616 cm². r = 88 ÷ (2 × 22/7) = 14, area = 22/7 × 196.
3. 125 cm³ and 150 cm². 5³; 6 × 25.
4. 24 cm² and 10 cm. ½ × 6 × 8; √(36 + 64).
5. 5 cm, 15π cm², 12π cm³. l = √(9 + 16); πrl = π × 3 × 5; ⅓ × π × 9 × 4.
6. 21%. 10 + 10 + 100/100 = 21, or 1.1² = 1.21.
7. 3,000 litres. Volume = 3 m³, and 1 m³ = 1,000 litres.
8. 72 cm². d²/2 = 144/2.

What to do next

• Copy both formula tables by hand once, then rewrite them from memory two days later.
• Do five path questions and five percentage-change questions this week.
• Squares up to 30 and cubes up to 15 make this chapter much faster; drill them with faster calculation.
• Percentage change in area uses the same successive-change idea as percentage.