In this guide
Mensuration means measuring shapes. You find a shape's boundary (perimeter), its surface (area) or the space inside it (volume). SSC GD keeps these questions practical: fencing a field, tiling a floor, painting a room, filling a tank.
You need about fifteen formulas and one habit: check the units before you multiply. Most wrong answers in this chapter come from mixing cm and m, or from using a diameter where the formula wants a radius.
Flat shapes (2D)
| Shape | Perimeter | Area |
|---|---|---|
| Square (side a) | 4a | a² |
| Rectangle (length l, breadth b) | 2(l + b) | l × b |
| Triangle (base b, height h) | sum of the three sides | ½ × b × h |
| Circle (radius r) | 2πr (circumference) | πr² |
Extra facts worth knowing:
- Diagonal of a square = a√2. Diagonal of a rectangle = √(l² + b²).
- In a right-angled triangle, hypotenuse² = base² + height² (Pythagoras). Common sets: 3, 4, 5 and 5, 12, 13 and 6, 8, 10.
- Diameter = 2 × radius. A wheel moves forward one circumference in one full turn.
Use π = 22/7 unless the question says 3.14. With 22/7, pick a radius that is a multiple of 7 and the 7s cancel.
Solid shapes (3D)
| Shape | Volume | Total surface area |
|---|---|---|
| Cube (side a) | a³ | 6a² |
| Cuboid (l, b, h) | l × b × h | 2(lb + bh + hl) |
| Cylinder (radius r, height h) | πr²h | 2πr(r + h) |
The curved surface of a cylinder alone is 2πrh. The area of the four walls of a room is 2(l + b) × h, which is the cuboid's surface without the floor and ceiling.
Units and litres
| Conversion | Value |
|---|---|
| 1 m | 100 cm |
| 1 m² | 10,000 cm² |
| 1 m³ | 10,00,000 cm³ |
| 1 litre | 1,000 cm³ |
| 1 m³ | 1,000 litres |
Solved examples
Example 1. A rectangular field is 40 m long and 25 m wide. Find its area and the cost of fencing it at ₹15 per metre.
- Area = 40 × 25 = 1,000 m².
- Perimeter = 2 × (40 + 25) = 130 m.
- Cost = 130 × 15 = ₹1,950.
Example 2. A wheel has a radius of 35 cm. How far does it travel in 100 full turns?
- Circumference = 2 × 22/7 × 35 = 220 cm.
- In 100 turns: 220 × 100 = 22,000 cm.
- 22,000 cm = 220 m.
Example 3. A right-angled triangle has a base of 6 cm and a height of 8 cm. Find its hypotenuse and area.
- Hypotenuse² = 36 + 64 = 100, so hypotenuse = 10 cm.
- Area = ½ × 6 × 8 = 24 cm².
Example 4. A room is 5 m long, 4 m wide and 3 m high. Find the cost of painting its four walls at ₹20 per m².
- Area of four walls = 2 × (5 + 4) × 3 = 54 m².
- Cost = 54 × 20 = ₹1,080.
Example 5. A cylindrical tank has a radius of 70 cm and a height of 2 m. How many litres does it hold?
- Change to metres: r = 0.7 m, h = 2 m.
- Volume = 22/7 × 0.7 × 0.7 × 2 = 3.08 m³.
- 3.08 × 1,000 = 3,080 litres.
Example 6. How many cubes of side 2 cm can be cut from a cube of side 6 cm?
- Big cube volume = 6³ = 216 cm³. Small cube volume = 2³ = 8 cm³.
- Number = 216 ÷ 8 = 27. (Check: 3 small cubes fit along each edge, and 3 × 3 × 3 = 27.)
What happens when sides change
| Change | Effect |
|---|---|
| Side of a square doubled | Area × 4 |
| Radius of a circle doubled | Area × 4, circumference × 2 |
| Side of a cube doubled | Volume × 8 |
| Radius of a cylinder doubled, same height | Volume × 4 |
Area grows with the square of the change and volume with the cube. This answers many "by how many times" questions without any formula work.
Common mistakes
| Mistake | Correct way |
|---|---|
| Using the diameter as r | Halve the diameter first |
| Mixing cm and m | Convert everything to one unit first |
| Taking 1 m² = 100 cm² | 1 m² = 10,000 cm² |
| Fencing cost on the area | Fencing uses perimeter; tiling uses area |
| Forgetting ½ in the triangle formula | Area = ½ × base × height |
Practice set
- The perimeter of a square is 64 m. Find its area.
- A rectangle has an area of 180 m² and a length of 15 m. Find its perimeter.
- The circumference of a circle is 88 cm. Find its area. (π = 22/7)
- Find the area of a triangle with a base of 12 cm and a height of 9 cm.
- Find the diagonal of a rectangle 12 cm by 5 cm.
- Find the volume and total surface area of a cube of side 5 cm.
- A tank is 3 m long, 2 m wide and 1.5 m deep. How many litres does it hold?
- Find the volume of a cylinder with a radius of 7 cm and a height of 10 cm.
Answers:
- Side = 64 ÷ 4 = 16 m. Area = 256 m².
- Breadth = 180 ÷ 15 = 12 m. Perimeter = 2 × 27 = 54 m.
- 2 × 22/7 × r = 88, so r = 14 cm. Area = 22/7 × 14 × 14 = 616 cm².
- ½ × 12 × 9 = 54 cm².
- √(144 + 25) = √169 = 13 cm.
- Volume = 125 cm³. Surface area = 6 × 25 = 150 cm².
- 3 × 2 × 1.5 = 9 m³ = 9,000 litres.
- 22/7 × 7 × 7 × 10 = 1,540 cm³.
What to do next
- Write the 2D and 3D tables from memory once a day for a week.
- Solve five fencing and five flooring questions, and write "perimeter" or "area" before each.
- Revise squares and square roots, which you need for diagonals and radius questions.
- Keep the formulas on the maths revision sheet for the final week.
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 Staff Selection Commission website .
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