Mensuration – Quick & Easy Study Notes
(For Haryana PSC – Quantitative Aptitude)
Goal: Know how to find the size (area, perimeter, volume, surface area) of common shapes.
Tip: Think of the shape as an everyday object. Use the word “size” instead of “area” or “volume”.
1. Why Mensuration Matters
- Real life: real estate, packaging, construction, sports.
- Exam: many short‑answer questions on area/perimeter/volume.
- Strategy: Memorise a cheat‑sheet of the most common formulas.
2. Basic 2‑D Shapes
| Shape | Everyday Analogy | Perimeter? (P) | Area? (A) |
|---|---|---|---|
| Square | A book cover | 4 × side |
side × side |
| Rectangle | A school board | 2 × (length + breadth) |
length × breadth |
| Triangle | A slice of pizza | sum of all sides |
½ × base × height |
| Parallelogram | A slanted box | 2 × (base + side) |
base × height |
| Trapezium (UK) / Trapezoid (US) | A table with one short side | sum of all sides |
½ × (sum of parallel sides) × height |
Tiny Example – Rectangle
- A book is 20 cm long and 12 cm wide.
- Perimeter =
2 × (20 + 12) = 64 cm. - Area =
20 × 12 = 240 cm².
3. Circle
| Concept | Analogy | Key Numbers |
|---|---|---|
Radius (r) |
Distance from centre to edge | |
Diameter (d) |
Full width through centre | d = 2 × r |
Circumference (C) |
Length of the edge | C = π × d ≈ 3.14 × d |
Area (A) |
Size inside the edge | A = π × r² ≈ 3.14 × r² |
Tiny Example – Circle
- A coin has a radius of 3 cm.
- Circumference =
π × 2 × 3 ≈ 18.84 cm. - Area =
π × 3² ≈ 28.26 cm².
4. Basic 3‑D Shapes
| Shape | Everyday Analogy | Volume (V) | Surface Area (S) |
|---|---|---|---|
| Cube | A dice | side³ |
6 × side² |
| Cuboid | A shoebox | length × breadth × height |
2 × (lb + bh + hl) |
| Sphere | A ball | (4/3) × π × r³ |
4 × π × r² |
| Cylinder | A can | π × r² × height |
2 × π × r × (r + height) |
| Cone | A party hat | (1/3) × π × r² × height |
π × r × (r + slant) (slant = √(r²+height²)) |
| Pyramid | A pyramid shape toy | (1/3) × base area × height |
base area + lateral area |
Remember: For all 3‑D shapes, volume is like “how much space inside”; surface area is like “how much paint needed to cover the outside”.
Tiny Example – Cylinder
- A water bottle: radius = 3 cm, height = 12 cm.
- Volume =
π × 3² × 12 ≈ 339.12 cm³. - Surface Area =
2 × π × 3 × (3 + 12) ≈ 2 × 3.14 × 3 × 15 ≈ 282.6 cm².
5. Quick Formula Cheat‑Sheet
| Concept | Formula | What to plug in |
|---|---|---|
| Square | P = 4a | a = side |
| A = a² | ||
| Rectangle | P = 2(l + b) | l = length, b = breadth |
| A = l × b | ||
| Triangle | P = a + b + c | a,b,c = sides |
| A = ½ × base × height | ||
| Circle | C = π × d | d = diameter |
| A = π × r² | r = radius |
|
| Cube | V = a³ | a = side |
| S = 6a² | ||
| Cuboid | V = l × b × h | |
| S = 2(lb + bh + hl) | ||
| Sphere | V = 4/3 π r³ | |
| S = 4 π r² | ||
| Cylinder | V = π r² h | |
| S = 2 π r (r + h) | ||
| Cone | V = 1/3 π r² h | |
| S = π r (r + √(r² + h²)) |
Tip: Use `