CUET PG – Quantitative Aptitude
Mensuration – Study Notes
Mensuration is the branch of mathematics that deals with the measurement of geometric figures such as lengths, areas, and volumes.
In CUET‑PG, you’ll encounter a mix of basic formulas and application questions that require quick substitutions and sometimes multiple steps.
1. Simple Explanation
-
Perimeter / Circumference – Total distance around a figure.
- Perimeter of a polygon = sum of all side lengths.
- Circumference of a circle = (2\pi r) (or (\pi d)).
-
Area – Measure of the surface covered by a figure.
- 2‑D shapes: use the “base × height” or “radius × radius” methods.
- 3‑D shapes: surface area = sum of areas of all faces.
-
Volume – Measure of space occupied by a solid.
- 3‑D shapes: usually “area of base × height” or “(\frac{1}{3}) × base area × height” for pyramids/cones.
2. Key Points & Formulas
| Shape | Parameters | Area | Perimeter / Circumference | Volume | Surface Area |
|---|---|---|---|---|---|
| Square | side = (s) | (s^2) | (4s) | – | – |
| Rectangle | length (l), breadth (b) | (lb) | (2(l+b)) | – | – |
| Triangle | base (b), height (h) | (\frac{1}{2}bh) | – | – | |
| Circle | radius (r) | (\pi r^2) | (2\pi r) | – | – |
| Parallelogram | base (b), height (h) | (bh) | – | – | |
| Trapezium | bases (a, b), height (h) | (\frac{1}{2}(a+b)h) | – | – | |
| Cube | side (a) | – | – | (a^3) | (6a^2) |
| Cuboid | length (l), breadth (b), height (h) | – | – | (lbh) | (2(lb+bh+hl)) |
| Cylinder | radius (r), height (h) | – | – | (\pi r^2 h) | (2\pi r(r+h)) |
| Cone | radius (r), height (h), slant height (l) | – | – | (\frac{1}{3}\pi r^2 h) | (\pi r(r+l)) |
| Sphere | radius (r) | – | – | (\frac{4}{3}\pi r^3) | (4\pi r^2) |
| Right Pyramid | base area (A_b), height (h) | – | – | (\frac{1}{3}A_b h) | – |
| Right Ellipsoid | semi‑axes (a,b,c) | – | – | (\frac{4}{3}\pi abc) | – (no simple closed form) |
Remember:
Use ( \pi = 3.14 ) or ( \pi = \frac{22}{7} ) as required by the question.
Always check the units (cm, m, etc.) – they usually cancel out in ratios.
3. Quick Revision Checklist
| Category | Formula | Key Notes |
|---|---|---|
| Perimeter | (P = 2(l+b)) (rect.) | Add all sides. |
| Circumference | (C = 2\pi r) | (r = \frac{d}{2}) |
| Area (2‑D) | (A = \frac{1}{2}bh) (tri.) | Use base × height. |
| Area (circle) | (A = \pi r^2) | (r = \frac{d}{2}) |
| Volume (cylinder) | (V = \pi r^2 h) | Base area × height. |
| Volume (cone) | (V = \frac{1}{3}\pi r^2 h) | 1/3 of cylinder. |
| Volume (sphere) | (V = \frac{4}{3}\pi r^3) | 4/3 of cylinder with same radius and height. |
| Surface Area (cylinder) | (S = 2\pi r(r+h)) | Lateral + top & bottom. |
| Surface Area (cone) | (S = \pi r(r+l)) | Lateral + base. |
| Surface Area (sphere) | (S = 4\pi r^2) | No lateral part. |
4. Worked Example
Problem (CUET‑PG style):
A right circular cone has a height of 12 cm and a slant height of 15 cm. Find
- the radius of the base
- the volume of the cone
- the total surface area of the cone.
Solution
Step 1 – Find base radius (r)
In a right cone, the slant height (l), radius (r), and height (h) satisfy the Pythagorean relation:
[ l^2 = r