In this guide
Mensuration questions in RRB Group D are direct: pick the right formula, put in the numbers, and do the arithmetic without a slip. The difficulty is not the idea. It is remembering which formula goes with which shape, and not losing marks to units or to π.
This guide gives you every formula you need in three tables, then shows the question types that appear most often: straight area and volume, reshaping a wire, melting one solid into another, and percentage change in area.
Flat shapes (2D)
| Shape | Perimeter | Area |
|---|---|---|
| Square (side a) | 4a | a² (or diagonal² ÷ 2) |
| Rectangle (length l, breadth b) | 2(l + b) | l × b |
| Triangle (base b, height h) | Sum of the three sides | ½ × b × h |
| Equilateral triangle (side a) | 3a | (√3/4) × a² |
| Parallelogram (base b, height h) | Sum of the sides | b × h |
| Rhombus (diagonals d₁, d₂) | 4 × side | ½ × d₁ × d₂ |
| Trapezium (parallel sides a, b; height h) | Sum of the sides | ½ × (a + b) × h |
| Circle (radius r) | 2πr | πr² |
| Semicircle (radius r) | πr + 2r | ½πr² |
Also useful: the diagonal of a rectangle is √(l² + b²), and the diagonal of a square is a√2.
Solid shapes (3D)
| Solid | Curved or lateral surface | Total surface | Volume |
|---|---|---|---|
| Cube (side a) | 4a² | 6a² | a³ |
| Cuboid (l, b, h) | 2h(l + b) | 2(lb + bh + hl) | l × b × h |
| Cylinder (r, h) | 2πrh | 2πr(r + h) | πr²h |
| Cone (r, h, slant l) | πrl | πr(l + r) | ⅓πr²h |
| Sphere (r) | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere (r) | 2πr² | 3πr² | (2/3)πr³ |
For a cone, the slant height is l = √(r² + h²). For a cube, the longest diagonal is a√3; for a cuboid it is √(l² + b² + h²).
Units
| Convert | Rule |
|---|---|
| Length | 1 m = 100 cm; 1 km = 1,000 m |
| Area | 1 m² = 10,000 cm²; 1 hectare = 10,000 m² |
| Volume | 1 m³ = 10,00,000 cm³ |
| Capacity | 1 litre = 1,000 cm³; 1 m³ = 1,000 litres |
Two ideas that save time
Reshaping and recasting. When a wire is bent into a new shape, the perimeter stays the same. When a solid is melted and recast, the volume stays the same. Write "old = new" and solve.
Percentage change in area. If every length is multiplied by k, the area is multiplied by k² and the volume by k³. So if each side goes up by 10%, the area goes up by 10 + 10 + (10 × 10)/100 = 21%, the same successive-change rule you use in percentage.
Worked examples
Example 1: A rectangle is 15 m by 10 m. Find its area and perimeter.
- Area = 15 × 10 = 150 m².
- Perimeter = 2 × (15 + 10) = 50 m.
Example 2: A circle has a diameter of 28 cm. Find its area and circumference.
- Radius = 14 cm.
- Area = 22/7 × 14 × 14 = 22 × 2 × 14 = 616 cm².
- Circumference = 2 × 22/7 × 14 = 88 cm.
Example 3: A cuboid is 8 cm × 5 cm × 3 cm. Find its volume and total surface area.
- Volume = 8 × 5 × 3 = 120 cm³.
- Surface area = 2 × (8 × 5 + 5 × 3 + 3 × 8) = 2 × (40 + 15 + 24) = 2 × 79 = 158 cm².
Example 4: A wire in the shape of a square of side 22 cm is bent into a circle. Find the radius of the circle.
- Length of wire = 4 × 22 = 88 cm. This becomes the circumference.
- 2 × 22/7 × r = 88, so r = 88 × 7 ÷ 44 = 14 cm.
- The circle's area is 616 cm², more than the square's 484 cm². For a fixed perimeter, a circle always encloses the most area.
Example 5: A metal sphere of radius 3 cm is melted and recast into a cylinder of radius 2 cm. Find the height of the cylinder.
- Volume of sphere = (4/3) × π × 27 = 36π cm³.
- Volume of cylinder = π × 2² × h = 4πh.
- 4πh = 36π, so h = 9 cm. The π cancels, so there is no need to use 22/7.
Example 6: A cylinder has radius 7 cm and height 12 cm. Find its volume in litres.
- Volume = 22/7 × 7 × 7 × 12 = 22 × 7 × 12 = 1,848 cm³.
- 1,000 cm³ = 1 litre, so the volume is 1.848 litres.
Common mistakes
- Using the diameter as the radius. Halve it first.
- Using total surface area when the question says "curved" (or the other way round). A pipe or a tent has no top and bottom, so only the curved surface counts.
- Forgetting the ⅓ in the cone's volume.
- Mixing metres and centimetres in the same formula.
- Taking the height of a cone as its slant height.
Practice set
- A square has a perimeter of 48 cm. Find its area.
- Find the area of a circle of radius 21 cm.
- Find the volume and total surface area of a cube of side 6 cm.
- Find the curved surface area of a cylinder with radius 14 cm and height 10 cm.
- Find the area of a triangle with base 12 cm and height 9 cm.
- The diagonals of a rhombus are 16 cm and 12 cm. Find its area and its side.
- A cone has radius 7 cm and height 24 cm. Find its slant height and curved surface area.
- A water tank is 2 m long, 1.5 m wide and 1 m deep. How many litres does it hold?
Answers:
- 144 cm². Side = 48 ÷ 4 = 12 cm.
- 1,386 cm². 22/7 × 21 × 21 = 22 × 3 × 21.
- 216 cm³ and 216 cm². 6³ = 216; 6 × 6² = 216.
- 880 cm². 2 × 22/7 × 14 × 10 = 2 × 22 × 2 × 10.
- 54 cm². ½ × 12 × 9.
- 96 cm² and 10 cm. Area = ½ × 16 × 12. The diagonals bisect each other at right angles, so the side is √(8² + 6²) = 10.
- 25 cm and 550 cm². √(49 + 576) = √625 = 25; 22/7 × 7 × 25 = 550.
- 3,000 litres. 2 × 1.5 × 1 = 3 m³, and 1 m³ = 1,000 litres.
What to do next
- Write the three tables from memory. Check them, and repeat until they are perfect.
- Solve five questions each on circles, cylinders and cones, and five on recasting.
- Mark every question that needed a unit conversion, and redo those after two days.
- Revise geometry for the Pythagoras triplets that hide in cone and rhombus questions.
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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