Karnataka TET – Quantitative Aptitude
Topic: Mensuration
(All explanations are kept very simple – think of everyday objects like a pizza, a coin, a box, etc.)
1. What is Mensuration?
Mensuration is the study of measuring the size of objects – length, width, height, area, and volume.
Think of it as “how to measure a shape or a solid”.
2. Basic Concepts
| Term | Everyday Analogy | Formula (if needed) |
|---|---|---|
| Perimeter | The length of a fence around a garden | Sum of all sides |
| Area | How many square tiles fit on a floor | Depends on shape |
| Circumference | The distance around a round table | (C = 2\pi r) |
| Radius | Distance from the center to the edge of a circle | (r) |
| Diameter | Distance across the center of a circle | (d = 2r) |
| Pi (π) | Roughly the ratio of a circle’s circumference to its diameter | ≈ 3.1416 |
| Volume | How many cubes fit inside a box | Depends on shape |
| Surface Area | Total area of all sides of a box | Depends on shape |
Tip: For every shape, first remember its area or volume formula, then multiply by π whenever a circle or sphere appears.
3. 2‑D Shapes (Area & Perimeter)
| Shape | Perimeter Formula | Area Formula | Simple Example |
|---|---|---|---|
| Rectangle | (P = 2(l + w)) | (A = l \times w) | A book: 15 cm × 20 cm → P = 70 cm, A = 300 cm² |
| Square | (P = 4a) | (A = a^2) | A chessboard: side 5 cm → P = 20 cm, A = 25 cm² |
| Triangle | (scalene) (P = a + b + c) | (any) (\frac{1}{2} \times \text{base} \times \text{height}) | A slice of pizza: base 6 cm, height 8 cm → A = 24 cm² |
| Circle | (C = 2\pi r) | (A = \pi r^2) | A coin: radius 1.5 cm → C ≈ 9.42 cm, A ≈ 7.07 cm² |
| Parallelogram | (P = 2(a + b)) | (A = \text{base} \times \text{height}) | A tilted box: base 10 cm, height 4 cm → A = 40 cm² |
| Trapezium (Trapezoid) | (P = a + b + c + d) | (A = \frac{1}{2}(a + b) \times h) | A roof shape: bases 8 cm & 12 cm, height 5 cm → A = 40 cm² |
Remember:
- Find the perimeter first if you need it.
- For triangles, you always need base × height / 2.
- For circles, replace radius by diameter/2 if you want to use (d).
4. 3‑D Shapes (Volume & Surface Area)
| Shape | Volume Formula | Surface Area (SA) Formula | Simple Example |
|---|---|---|---|
| Cube | (V = a^3) | (SA = 6a^2) | A small box: side 4 cm → V = 64 cm³, SA = 96 cm² |
| Rectangular Prism (Cuboid) | (V = lwh) | (SA = 2(lw + lh + wh)) | A shoebox: 10 × 5 × 2 cm → V = 100 cm³, SA = 140 cm² |
| Cylinder | (V = \pi r^2 h) | (SA = 2\pi r(h + r)) | A can: r = 3 cm, h = 7 cm → V ≈ 197 cm³, SA ≈ 188 cm² |
| Sphere | (V = \frac{4}{3}\pi r^3) | (SA = 4\pi r^2) | A ball: r = 5 cm → V ≈ 523.6 cm³, SA ≈ 314 cm² |
| Cone | (V = \frac{1}{3}\pi r^2 h) | (SA = \pi r(r + l)) (where (l) = slant height) | A party hat: r = 2 cm, h = 4 cm → V ≈ 33.5 cm³ |
Quick Check:
- For a cylinder, imagine a can of soup.
- For a cone, think of a pizza box.
- For a sphere, picture a ball.
5. Common Mistakes to Avoid
| Mistake | What to Do Instead |
|---|---|
| Mixing up diameter and radius | Always check if the problem gives diameter (d); then (r = d/2). |
| Forgetting the factor ½ for triangles | Always divide base × height by 2. |
| Using π ≈ 3.14 only when needed | Use 3.14 for quick mental math; if the question asks for exact value, keep π as is. |
| Not squaring units | Remember that area is in square units (cm²) and volume in cubic units (cm³). |
6. Tiny Example Problems
Problem 1: Circle
Question: A round plate has a diameter of 20 cm. Find its area.
Solution:
- Radius (r = d/2 = 10) cm.
- Area (A = \pi