In this guide
Mensuration is about measuring shapes: how much fence goes around a field, how much paint covers a tank, how much water it holds. RRB NTPC questions usually need one formula, a sensible choice of π and careful arithmetic.
The formulas are the foundation, so they are gathered in two tables below. After that come the ideas that turn formula-recall into quick solving: why the formulas work, how shapes scale, and what stays the same when a solid is melted and recast. Brush up on geometry first if triplets and circle facts feel rusty.
2D shapes
| Shape | Perimeter | Area |
|---|---|---|
| Square (side a) | 4a | a²; also diagonal² ÷ 2 |
| Rectangle (l, b) | 2(l + b) | l × b |
| Triangle (base b, height h) | Sum of sides | ½ × b × h |
| Triangle (sides a, b, c) | a + b + c | √(s(s − a)(s − b)(s − c)), s = half the perimeter |
| Equilateral triangle (side a) | 3a | (√3/4) × a² |
| Rhombus (diagonals d₁, d₂) | 4 × side | ½ × d₁ × d₂ |
| Trapezium (parallel sides a, b; height h) | Sum of sides | ½ × (a + b) × h |
| Circle (radius r) | 2πr | πr² |
| Semicircle (radius r) | πr + 2r | ½πr² |
3D shapes
| Solid | Surface area | Volume |
|---|---|---|
| Cube (side a) | 6a² | a³ |
| Cuboid (l, b, h) | 2(lb + bh + hl) | l × b × h |
| Cylinder (r, h) | Curved 2πrh; total 2πr(r + h) | πr²h |
| Cone (r, h, slant l) | Curved πrl; total πr(l + r) | ⅓πr²h |
| Sphere (r) | 4πr² | (4/3)πr³ |
| Hemisphere (r) | Curved 2πr²; total 3πr² | (2/3)πr³ |
For a cone, slant height l = √(r² + h²), from the right triangle formed by the radius, the height and the slant.
Why the main formulas work
- Prisms and cylinders: volume = area of the base × height. A cylinder is a stack of circles, so πr² × h.
- Cone: a cone holds exactly one-third of a cylinder with the same base and height, hence ⅓πr²h.
- Curved surface of a cylinder: unroll it and you get a rectangle whose length is the circumference 2πr and whose width is h.
- Hemisphere total: curved part 2πr² plus the flat circular top πr² gives 3πr².
Choosing π
- If the radius or diameter is a multiple of 7, use π = 22/7. The 7 cancels and the arithmetic stays whole.
- If the options contain π (such as 36π cm³), leave π as it is.
- Use 3.14 only when the question says so.
Scaling: when every length changes
If every length is multiplied by k, perimeter × k, area × k², volume × k³.
Why: area multiplies two lengths together and volume multiplies three. So a 20% increase in radius (k = 1.2) raises area by a factor of 1.44, which is a 44% increase, not 40%.
Melting and recasting
When a solid is melted and made into other shapes, volume stays the same. Surface area does not.
- Number of small solids = volume of big solid ÷ volume of one small solid.
- For spheres, this reduces to (R ÷ r)³, because the 4/3 and π cancel.
Units
- 1 m = 100 cm, so 1 m² = 10,000 cm² and 1 m³ = 10,00,000 cm³.
- 1 litre = 1,000 cm³, and 1 m³ = 1,000 litres.
- 1 hectare = 10,000 m².
Worked questions at NTPC level
Q1. Find the area and circumference of a circle of radius 7 cm.
Area = 22/7 × 49 = 154 cm². Circumference = 2 × 22/7 × 7 = 44 cm.
Q2. Find the area of a triangle with sides 13 cm, 14 cm and 15 cm.
s = 21. Area = √(21 × 8 × 7 × 6) = √7,056 = 84 cm².
Q3. Find the volume and curved surface area of a cylinder with radius 7 cm and height 10 cm.
Volume = 22/7 × 49 × 10 = 1,540 cm³.
Curved surface area = 2 × 22/7 × 7 × 10 = 440 cm².
Q4. A cone has radius 7 cm and height 24 cm. Find its slant height, curved surface area and volume.
l = √(49 + 576) = √625 = 25 cm (a 7-24-25 triplet).
Curved surface area = 22/7 × 7 × 25 = 550 cm².
Volume = ⅓ × 22/7 × 49 × 24 = 1,232 cm³.
Q5. A metal sphere of radius 6 cm is melted and recast into small spheres of radius 2 cm. How many small spheres are made?
(6 ÷ 2)³ = 27.
Q6. The radius of a circle is increased by 20%. By what percentage does its area increase?
k = 1.2, so area × 1.44. Increase = 44%.
Common mistakes
- Using the diameter as the radius. Halve it first.
- Converting lengths but not areas or volumes. 1 m² is 10,000 cm², not 100.
- Giving curved surface area when total is asked, or the other way round.
- Using height instead of slant height in the cone's surface area.
- Adding percentages in scaling. A 20% rise in radius is a 44% rise in area.
Practice set
- Find the area of a square of side 9 cm.
- Find the area of a circle of radius 14 cm.
- Find the volume of a cuboid 10 cm × 6 cm × 4 cm.
- Find the volume of a cylinder with radius 7 cm and height 5 cm.
- The circumference of a circle is 88 cm. Find its radius.
- The diagonals of a rhombus are 16 cm and 12 cm. Find its area and side.
- A water tank measures 2 m × 1.5 m × 1 m. How many litres does it hold?
- Find the surface area of a sphere of radius 21 cm.
Answers:
- 81 cm². 9 × 9.
- 616 cm². 22/7 × 196.
- 240 cm³. 10 × 6 × 4.
- 770 cm³. 22/7 × 49 × 5.
- 14 cm. 2 × 22/7 × r = 88.
- Area 96 cm², side 10 cm. ½ × 16 × 12; the half-diagonals 8 and 6 give side 10.
- 3,000 litres. 3 m³ × 1,000.
- 5,544 cm². 4 × 22/7 × 441.
What to do next
- Copy both formula tables by hand, then rewrite them from memory until they come out correct.
- Solve 20 questions, circling whether each needs perimeter, area, surface area or volume before you start.
- Practise five scaling and five melting-recasting questions.
- Revise percentages for the "increased by x%" variety.
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 Railway Recruitment Boards website .
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