📐 Mensuration – Quick & Easy Study Notes
(For Madhya Pradesh Patwari / Lekhpal Quantitative Aptitude)
What is Mensuration?
It’s the branch of math that helps you find the size of things – how big a shape is (area), how long its edges are (perimeter), and how much space it takes up (volume). Think of it like measuring a pizza, a rectangular garden, or a box.
1️⃣ 2‑D Shapes (Flat Shapes)
| Shape | Everyday Analogy | What We Want to Know | Simple Formula |
|---|---|---|---|
| Rectangle / Square | A pizza slicer (long & short sides) | Area & Perimeter | Area = Length × Breadth Perimeter = 2 × (Length + Breadth) |
| Triangle | A slice of pizza | Area & Perimeter | Area = ½ × Base × Height Perimeter = a + b + c |
| Circle | A round pizza | Area & Circumference | Area = π × Radius² Circumference = 2 × π × Radius |
Tip:
- π (Pi) ≈ 3.14 (use 3.14 if not given).
- If you know the diameter (full width), radius =
Diameter ÷ 2.
2️⃣ 3‑D Shapes (Solid Shapes)
| Shape | Everyday Analogy | What We Want to Know | Simple Formula |
|---|---|---|---|
| Cube | A box of chocolates | Surface Area & Volume | Surface Area = 6 × Side² Volume = Side³ |
| Cuboid / Rectangular Prism | A shoebox | Surface Area & Volume | Surface Area = 2(lw + lh + wh) Volume = Length × Width × Height |
| Cylinder | A milk jug | Surface Area & Volume | Surface Area = 2πr(r + h) Volume = πr²h |
| Sphere | A ball | Surface Area & Volume | Surface Area = 4πr² Volume = (4/3)πr³ |
| Cone | A party hat | Surface Area & Volume | Surface Area = πr(r + l) Volume = (1/3)πr²h (where l = slant height) |
Quick Memory:
- Cube: all sides equal →
6 × side²for area.- Cylinder: imagine a can – top + bottom + side.
3️⃣ Quick Calculation Tricks
| Trick | When to Use | Example |
|---|---|---|
| Use ½ for triangles | Right‑angled or any triangle | Triangle with base 8 m & height 5 m → ½ × 8 × 5 = 20 m² |
| π ≈ 3 | Rough estimate | Circle with radius 4 m → 3 × 4² = 48 m² |
| Half the area for a right‑angled triangle | Right‑angled only | 10 m × 6 m → ½ × 10 × 6 = 30 m² |
| Double the radius for diameter | Needed in circle formulas | Radius 7 m → Diameter = 2 × 7 = 14 m |
4️⃣ Tiny Example Problems
Example 1 – Rectangle
Problem: A field is 20 m long and 15 m wide.
Find: Area.
Solution:Area = 20 × 15 = 300 m².
Example 2 – Triangle
Problem: A triangular plot has base 12 m and height 9 m.
Find: Area.
Solution:Area = ½ × 12 × 9 = 54 m².
Example 3 – Circle
Problem: A circular pond has a radius of 5 m.
Find: Circumference.
Solution:Circumference = 2 × π × 5 ≈ 2 × 3.14 × 5 ≈ 31.4 m.
Example 4 – Cylinder
Problem: A milk jug has radius 4 cm and height 10 cm.
Find: Volume.
Solution:Volume = π × 4² × 10 ≈ 3.14 × 16 × 10 ≈ 502.4 cm³.
Example 5 – Cube
Problem: A cube box has side length 6 cm.
Find: Surface Area.
Solution:Surface Area = 6 × 6² = 6 × 36 = 216 cm².
5️⃣ Quick Reference Sheet
| Shape | Area Formula | Surface Area (if 3‑D) | Volume |
|---|---|---|---|
| Rectangle | L × B |
– | – |
| Triangle | ½ × Base × Height |
– | – |
| Circle | πr² |
– | – |
| Square | Side² |
– | – |
| Cube | – | 6 × Side² |
Side³ |
| Cuboid | – | 2(lw + lh + wh) |
L × W × H |
| Cylinder | – | 2πr(r + h) |
πr²h |
| Sphere | – | 4πr² |
(4/3)πr³ |
| Cone | – | πr(r + l) |
(1/3)πr²h |
📌 Key Takeaways
- Area = “how much floor space?”
- Perimeter / Circumference = “total edge length.”
- Volume = “how much space inside?”
- Use π = 3.14 (or 3 for quick checks).
- Remember ½