Mensuration – Study Notes (TOPIK Quantitative Aptitude)
Mensuration is the branch of mathematics that deals with the measurement of geometric figures – area, perimeter, volume, surface area, and related quantities.
1. Simple Explanation
| Concept | What It Means |
|---|---|
| Area | Size of a 2‑D shape’s interior (square units). |
| Perimeter | Total length of a shape’s boundary (linear units). |
| Volume | Amount of space a 3‑D object occupies (cubic units). |
| Surface Area | Total area of all surfaces of a 3‑D object (square units). |
| Radius/Radius | Distance from center to any point on a circle/ sphere. |
| Diameter | 2 × radius. |
| Height | Distance from base to top in a prism, cylinder, or cone. |
2. Key Points & Formulas
| Shape | Formula | Notes |
|---|---|---|
| Rectangle | (A = l \times w) (P = 2(l + w)) |
(l) = length, (w) = width |
| Square | (A = s^2) (P = 4s) |
All sides equal |
| Triangle | (A = \frac{1}{2} b h) (P = a + b + c) |
(b) = base, (h) = height |
| Parallelogram | (A = b \times h) | Same as rectangle but height is perpendicular to base |
| Trapezium / Trapezoid | (A = \frac{h}{2} (a + b)) | (a, b) = parallel sides |
| Circle | (A = \pi r^2) (C = 2\pi r) |
(r) = radius |
| Cylinder | (V = \pi r^2 h) (SA = 2\pi r (r + h)) |
Two circles + side surface |
| Sphere | (V = \frac{4}{3}\pi r^3) (SA = 4\pi r^2) |
Only one radius |
| Cone | (V = \frac{1}{3}\pi r^2 h) (SA = \pi r (r + l)) |
(l) = slant height |
| Right Pyramid | (V = \frac{1}{3} B h) (SA = B + \frac{1}{2} P l) |
(B) = base area, (P) = perimeter of base, (l) = slant height |
| Cuboid (Rectangular Prism) | (V = lwh) (SA = 2(lw + lh + wh)) |
Common Constants
- (\pi \approx 3.1416)
- Use (22/7) for quick mental calculations.
3. Worked Example
Problem
A right circular cone has a radius of 6 cm and a slant height of 10 cm.
Find:
- The height of the cone.
- The volume.
- The total surface area.
Step 1: Find the Height
Use the Pythagorean theorem in the cone’s cross‑section:
[ l^2 = r^2 + h^2 \quad \Rightarrow \quad h = \sqrt{l^2 - r^2} ]
[ h = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 \text{ cm} ]
Step 2: Volume
[ V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (6)^2 (8) ]
[ V = \frac{1}{3}\pi \times 36 \times 8 = \frac{1}{3}\pi \times 288 = 96\pi \text{ cm}^3 ]
[ V \approx 96 \times 3.1416 \approx 301.6 \text{ cm}^3 ]
Step 3: Surface Area
[ SA = \pi r (r + l) = \pi \times 6 (6 + 10) ]
[ SA = \pi \times 6 \times 16 = 96\pi \text{ cm}^2 ]
[ SA \approx 96 \times 3.1416 \approx 301.6 \text{ cm}^2 ]
Answer
- Height = 8 cm
- Volume = (96\pi \approx 301.6) cm³
- Surface Area = (96\pi \approx 301.6) cm²
4. Quick Revision Summary
| Concept | Remember |
|---|---|
| Area of circle | (\pi r^2) |
| Circumference | (2\pi r) |
| Volume of cylinder | (\pi r^2 h) |
| Volume of cone | (\frac{1}{3}\pi r^2 h) |
| Surface area of sphere | (4\pi r^2) |
| Surface area of cylinder | (2\pi r(r + h)) |
| Volume of cuboid | (lwh) |
| Surface area of cuboid | (2(lw + lh + wh)) |
| Use Pythagoras | For slant height → (l = \sqrt{r^2 + h^2}) |
| Common shortcuts | (\pi \approx 22/7) or 3.14 for mental math |