Mensuration – Study Notes for AP Police SI (Quantitative Aptitude)
Mensuration is the branch of mathematics that deals with the measurement of geometric figures (length, area, volume, surface area, etc.). It is a core part of the Quantitative Aptitude section of the Andhra Pradesh Police Sub‑Inspector exam.
1. Simple Explanation
- What it is: Calculating the size of geometric shapes – e.g., how long the boundary (perimeter) is, how big the inside (area) is, or how much space a solid occupies (volume).
- Why it matters: Many SI questions ask you to find the area of a plot, the surface of a cylindrical can, or the volume of a water tank.
- How to approach:
- Identify the figure (triangle, rectangle, circle, cylinder, cone, sphere, etc.).
- Pick the right formula(s).
- Plug in the given values, do the arithmetic carefully.
- Check for units (cm², m³, etc.) if required.
2. Key Points & Formulas
| Figure | Perimeter / Circumference | Area | Surface Area | Volume |
|---|---|---|---|---|
| Line & Segment | – | – | – | – |
| Triangle | – | ( \frac{1}{2} \times \text{base} \times \text{height} ) | – | – |
| – | ( \sqrt{s(s-a)(s-b)(s-c)} ) (Heron) | – | – | |
| Rectangle / Square | (2(l+w)) | (l \times w) | – | – |
| Parallelogram | (2(a+b)) | (\text{base} \times \text{height}) | – | – |
| Trapezium | – | (\frac{1}{2}(a+b)\times h) | – | – |
| Circle | (2\pi r) | (\pi r^{2}) | – | – |
| Sphere | – | – | (4\pi r^{2}) | (\frac{4}{3}\pi r^{3}) |
| Cylinder | – | – | (2\pi r(r+h)) | (\pi r^{2}h) |
| Cone | – | – | (\pi r(r+l)) (l = slant height) | (\frac{1}{3}\pi r^{2}h) |
| Pyramid | – | – | (A_{\text{base}} + \frac{1}{2} P_{\text{base}} \times l) | (\frac{1}{3} A_{\text{base}} h) |
Quick Reference for Common Shapes
| Shape | Area | Surface Area | Volume |
|---|---|---|---|
| Triangle | ( \frac{1}{2} b h ) | – | – |
| Rectangle | (lw) | – | – |
| Circle | (\pi r^2) | – | – |
| Cylinder | – | (2\pi r (r+h)) | (\pi r^2 h) |
| Cone | – | (\pi r (r+l)) | (\frac{1}{3}\pi r^2 h) |
| Sphere | – | (4\pi r^2) | (\frac{4}{3}\pi r^3) |
Tip: For any 3‑D shape, surface area = sum of all outer faces; volume = base area × height (or the appropriate formula).
3. Common Mistakes to Avoid
| Mistake | What to watch out for | How to correct |
|---|---|---|
| Mixing r (radius) and a (diameter) | Forgetting that (d = 2r) | Always check the question for “diameter” or “radius.” |
| Using π = 3.14 vs π = 22/7 | Different problems may expect a specific value | Read the instruction; if none, use (22/7) for quick mental math. |
| Forgetting to square the radius in area | ( \pi r^2) vs ( \pi r) | Double‑check the formula. |
| Misapplying Heron’s formula | Wrong side lengths or wrong (s) | Compute (s = (a+b+c)/2) first, then plug in. |
| Ignoring unit conversions | e.g., cm to m | Always convert units before computing or after, but consistently. |
4. Worked Example
Problem:
A cylindrical water tank has a radius of 5 m and a height of 12 m. What is its volume? (Take π = 22/7)
Solution Steps
- Identify: Cylinder → formula for volume ( V = \pi r^{2} h ).
- Insert values:
[ V = \frac{22}{7} \times (5)^{2} \times 12 ] - Compute:
- (5^{2} = 25)
- ( \frac{22}{7} \times 25 = \frac{550}{7} \approx 78.571 )
- Multiply by 12:
[ 78.571 \times 12 \approx 942.857 \text{ m}^{3} ]
- Answer: The tank can hold **≈