📐 Mensuration – Quick‑Study Notes
(AWS Certification – Quantitative Aptitude)
Mensuration is the branch of mathematics that deals with the measures (area, volume, surface area, etc.) of geometric figures.
📚 Table of Contents
| Section | Description |
|---|---|
| 1. Simple Explanation | What mensuration is and why it matters |
| 2. Key Formulas & Concepts | 2‑D → 3‑D, composite shapes, special solids |
| 3. Worked Example | Step‑by‑step calculation |
| 4. Quick Revision Summary | One‑page cheat sheet |
| 5. Extra Practice | Sample questions to test your recall |
1️⃣ Simple Explanation
- Goal: Convert the dimensions of a shape (length, width, height, radius, etc.) into a numeric measure (area, volume, surface area).
- Why AWS cares: Many certification questions test problem‑solving skills that involve geometry, helping you think logically and apply formulas efficiently.
2️⃣ Key Points & Formulas
| Shape | 2‑D Measure | 3‑D Measure | Surface Area (SA) | Notes |
|---|---|---|---|---|
| Rectangle | (A = l \times w) | — | — | |
| Square | (A = a^2) | — | — | |
| Triangle | (A = \frac{1}{2} b h) | — | — | |
| Circle | (A = \pi r^2) | — | — | |
| Sphere | — | (V = \frac{4}{3}\pi r^3) | (SA = 4\pi r^2) | |
| Cube | — | (V = a^3) | (SA = 6a^2) | |
| Cuboid | — | (V = lwh) | (SA = 2(lw + lh + wh)) | |
| Cylinder | — | (V = \pi r^2 h) | (SA = 2\pi r(h + r)) | |
| Cone | — | (V = \frac{1}{3}\pi r^2 h) | (SA = \pi r(r + \sqrt{r^2 + h^2})) | Lateral area (=\pi r \ell) |
| Pyramid | — | (V = \frac{1}{3} B h) | (SA = B + \frac{1}{2} P \ell) | (B)=base area, (P)=perimeter of base, (\ell)=slant height |
| Frustum of a Cone | — | (V = \frac{1}{3}\pi h (R^2 + Rr + r^2)) | (SA = \pi (R + r) \ell + \pi R^2 + \pi r^2) |
Common Derived Quantities
| Quantity | Formula | Example Use |
|---|---|---|
| Circumference | (C = 2\pi r) | Needed for SA of cylinder/cone |
| Slant Height | (\ell = \sqrt{r^2 + h^2}) (cone) | Needed for SA of cone |
| Mean Radius | (\frac{R + r}{2}) (frustum) | For SA and volume of frustum |
Tips for Quick Calculations
- Keep π as 22/7 for mental math unless higher precision is required.
- Unit‑conversion first: If dimensions are in cm, keep them throughout; you’ll get cm² or cm³.
- Break composite shapes into simpler solids, compute each, then combine.
- Check units: Area → unit²; Volume → unit³; Surface area → unit².
3️⃣ Worked Example
Problem:
A right circular cylinder has a radius of 5 cm and a height of 12 cm.
Find
a) Volume
b) Surface area (including top & bottom)
| Step | Calculation | Result |
|---|---|---|
| a) Volume | (V = \pi r^2 h) | (V = \pi (5)^2 (12) = 300\pi \text{ cm}^3 \approx 942.48\ \text{cm}^3) |
| b) Surface area | (SA = 2\pi r (h + r)) | (SA = 2\pi (5)(12 + 5) = 2\pi(5)(17) = 170\pi \text{ cm}^2 \approx 534.07\ \text{cm}^2) |
Check:
- The lateral area (= 2\pi r h = 2π(5)(12) = 120π).
- The top & bottom area (= 2π r^2 = 2π(25) = 50π).
- Sum (= 120π + 50π = 170π). ✔️
4️⃣ Quick Revision Summary (Cheat Sheet)
| Shape | Volume | Surface Area |
|---|---|---|
| Cube | (a^3) | (6a^2) |
| Cuboid | (lwh) | (2(lw + lh + wh)) |
| Sphere | (\frac{4}{3}\pi r^3) | (4\pi r^2) |
| Cylinder | (\pi r^2 h) | ( |