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Mensuration for IBPS PO

Areas and perimeters, cylinders, cones and spheres, a sphere melted into a cylinder, a path around a park, and what happens to an area when the sides change. The formulas and the reasons behind the shortcuts, with six worked examples and eight practice questions for IBPS PO.

13 Oct 2026 6 min read

In this guide
  1. 2D shapes
  2. 3D shapes
  3. The shortcuts and why they work
  4. Worked examples
  5. Common mistakes
  6. Practice set
  7. What to do next

Mensuration is one of the most formula-dependent topics in IBPS PO quant, which also makes it one of the most learnable. The questions do not ask for tricks. They ask whether you know the right formula, pick the right radius, and keep units straight.

The topic appears as standalone arithmetic in the prelims, as statements in quantity comparison, where two shapes are compared, and occasionally inside data sufficiency. Use π = 22/7 unless the question says otherwise; the numbers are usually chosen so that 7 cancels.

2D shapes

ShapeAreaPerimeter and other facts
Rectangle (l, b)l × b2(l + b); diagonal √(l² + b²)
Square (side a)a²4a; diagonal a√2; area = diagonal² ÷ 2
Circle (radius r)πr²Circumference 2πr
Semicircleπr² ÷ 2Perimeter πr + 2r
Triangle½ × base × heightSum of sides
Equilateral triangle (side a)(√3/4) × a²Height (√3/2) × a
Right triangle½ × product of the two shorter sidesHypotenuse² = sum of squares of the other two
Trapezium½ × (sum of parallel sides) × height
Rhombus½ × d1 × d2Side = ½√(d1² + d2²)

3D shapes

SolidVolumeSurface area
Cuboid (l, b, h)lbh2(lb + bh + hl); diagonal √(l² + b² + h²)
Cube (side a)a³6a²
Cylinder (r, h)πr²hCurved 2πrh; total 2πr(r + h)
Cone (r, h, slant l)⅓πr²hCurved πrl; total πr(r + l); l = √(r² + h²)
Sphere (r)(4/3)πr³4πr²
Hemisphere (r)(2/3)πr³Curved 2πr²; total 3πr²

Why a cone is one-third of a cylinder: a cone with the same base and height as a cylinder holds exactly one-third of the cylinder's volume. That is why ⅓ appears in the cone formula and nowhere else in the table.

The shortcuts and why they work

  • Percentage change in area. If the length changes by a% and the breadth by b%, the area changes by a + b + ab/100 per cent. Why: area is a product, so the two changes act one after the other, exactly like successive percentage changes.
  • Scaling. If every length is multiplied by k, area is multiplied by k² and volume by k³. Why: area has two length dimensions and volume has three. Double a sphere's radius and its volume becomes 8 times.
  • Melting and recasting. Volume stays the same; the shape changes. Set old volume = new volume (multiplied by the number of pieces, if any).
  • Wheels. One full turn covers one circumference. Revolutions = distance ÷ (2πr).
  • Units. 1 m³ = 1,000 litres. 1 hectare = 10,000 m².

Worked examples

Example 1 (rectangle): A rectangle is 24 m by 18 m. Find its area and diagonal.

  • Area = 24 × 18 = 432 m².
  • Diagonal = √(576 + 324) = √900 = 30 m.

Example 2 (circle from area): A circle has an area of 1,386 cm². Find its circumference.

  • r² = 1,386 × 7/22 = 441, so r = 21 cm.
  • Circumference = 2 × 22/7 × 21 = 132 cm.

Example 3 (cylinder): A cylinder has radius 7 cm and height 20 cm. Find its volume and total surface area.

  • Volume = 22/7 × 7 × 7 × 20 = 3,080 cm³.
  • Total surface area = 2 × 22/7 × 7 × (7 + 20) = 44 × 27 = 1,188 cm².

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

  • Sphere volume = (4/3) × π × 216 = 288π.
  • Cylinder volume = π × 16 × h.
  • 16h = 288, so h = 18 cm. π cancels, so there is no need to use 22/7 at all.

Example 5 (path around a park): A rectangular park is 40 m by 30 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.

  • Outer rectangle: (40 + 5) × (30 + 5) = 45 × 35 = 1,575 m². The width is added on both sides.
  • Park: 40 × 30 = 1,200 m².
  • Path = 375 m². Cost = 375 × 20 = ₹7,500.

Example 6 (percentage change): The length of a rectangle rises by 20% and its breadth falls by 20%. What happens to the area?

  • 20 + (−20) + (20 × −20)/100 = −4. The area falls by 4%.
  • Check: a 10 × 10 rectangle becomes 12 × 8 = 96, which is 4% less than 100.

Common mistakes

  • Using the diameter as the radius. Halve it first.
  • Mixing the height and the slant height of a cone. Volume uses h; curved surface uses l.
  • Adding the path width only once. A path around a field adds the width on both sides of each dimension.
  • Forgetting the ⅓ for a cone or the 4/3 for a sphere.
  • Adding percentage changes directly. Use a + b + ab/100.
  • Mixed units. Convert cm to m (or the reverse) before calculating. Area conversions square the factor: 1 m² = 10,000 cm².

Practice set

  1. A square has a diagonal of 10√2 cm. Find its area.
  2. A circle has a circumference of 88 cm. Find its area.
  3. Find the volume and total surface area of a cuboid measuring 12 cm × 8 cm × 5 cm.
  4. Find the volume of a cylinder with radius 14 cm and height 10 cm.
  5. The length and breadth of a rectangle both rise by 10%. By what percentage does the area rise?
  6. A wheel has a radius of 35 cm. How many revolutions does it make to cover 1.1 km?
  7. A cone has radius 6 cm and height 8 cm. Find its slant height, curved surface area and volume in terms of π.
  8. A tank measures 20 m × 15 m × 4 m. How many litres of water can it hold?

Answers:

  1. 100 cm². Area = diagonal² ÷ 2 = 200 ÷ 2. (The side is 10 cm.)
  2. 616 cm². 2 × 22/7 × r = 88 gives r = 14; area = 22/7 × 196.
  3. 480 cm³ and 392 cm². 12 × 8 × 5; 2(96 + 40 + 60) = 2 × 196.
  4. 6,160 cm³. 22/7 × 196 × 10.
  5. 21%. 10 + 10 + 100/100.
  6. 500. Circumference = 2 × 22/7 × 35 = 220 cm; 1.1 km = 1,10,000 cm; 1,10,000 ÷ 220 = 500.
  7. Slant 10 cm, curved surface 60π cm², volume 96π cm³. l = √(36 + 64); πrl = π × 6 × 10; ⅓ × π × 36 × 8.
  8. 12,00,000 litres. Volume = 1,200 m³, and 1 m³ = 1,000 litres.

What to do next

  • Write both formula tables from memory twice this week, including the hemisphere and cone surface areas.
  • Solve eight mensuration questions a day for five days, writing the formula before each.
  • Revise percentage for the change-in-area rule, and try mensuration statements inside quantity comparison.

A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Institute of Banking Personnel Selection website .

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