Geometry Basics – Quantitative Aptitude
For the Military Nursing Service (MNS) Entrance Test
1. Simple Explanation
Geometry in quantitative aptitude deals with the properties of shapes, sizes, and relative positions of points, lines, angles, planes, and solids.
- Points – location with no size.
- Lines – straight path extending infinitely in both directions (or a segment).
- Planes – flat, two‑dimensional surfaces.
- Angles – measure of rotation between two intersecting lines.
- Shapes – polygons (triangles, quadrilaterals, etc.) and circles.
- Area & Perimeter – size of a shape’s surface and length around it.
- Pythagoras – relationship in right triangles.
These concepts allow you to solve quick arithmetic problems such as “What is the area of a triangle?” or “How many edges does a cube have?”
2. Key Points & Formulas
| Category | Symbol | Formula | Quick Tip |
|---|---|---|---|
| Basic Shapes | Triangle | – | Use base × height ÷ 2 |
| Rectangle | – | Length × Breadth | |
| Square | – | Side² | |
| Parallelogram | – | Base × Height | |
| Trapezium | – | (a + b) × Height ÷ 2 | |
| Circle | – | Area πr², Circumference 2πr | |
| Angles | Sum of angles in triangle | 180° | If two angles known, third = 180° – sum |
| Complementary | 90° – α | Complementary + α = 90° | |
| Supplementary | 180° – α | Supplementary + α = 180° | |
| Pythagorean Theorem | Right triangle | (a^2 + b^2 = c^2) | For hypotenuse: (c = √(a²+b²)) |
| Area of Regular Polygons | Regular n‑gon | (\frac{n \times s^2}{4\tan(\pi/n)}) | For quick problems, use known values: 5‑gon 1/4, 6‑gon 1/3, etc. |
| Volume (3‑D) | Cube | (s^3) | |
| Rectangular Prism | (l \times w \times h) | ||
| Cylinder | (\pi r^2 h) | ||
| Sphere | (\frac{4}{3}\pi r^3) | ||
| Conversion | Degrees ↔ Radians | (π) radians = 180° | Remember 180° = π rad |
Common Trigonometric Ratios (Right Triangle)
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
3. Worked Example – “Find the area of a triangle with base 12 cm and height 8 cm”
Step 1: Identify the formula
Area of a triangle = ½ × base × height
Step 2: Plug in the values
Area = ½ × 12 cm × 8 cm
= 0.5 × 96 cm²
= 48 cm²
Result: The area of the triangle is 48 cm².
Quick Check:
- If you drop the ½, you get 96 – that would be the product of base and height, not area.
- Remember: “area = ½ × base × height” is the most common geometry formula in aptitude tests.
4. Quick Revision Summary
| Concept | Formula / Rule | Use in Aptitude |
|---|---|---|
| Area of Triangle | ½ × base × height | Most common area problem |
| Area of Rectangle/Square | L × B or s² | Easy multiplication |
| Perimeter | Sum of all sides | Quick addition |
| Circle | A = πr², C = 2πr | Know π ≈ 3.14 |
| Pythagoras | a²+b²=c² | For right‑angle questions |
| Angle Sum | Triangle = 180°, Quadrilateral = 360° | Solve missing angles |
| Complementary/Supplementary | α+β = 90° or 180° | Quick angle checks |
| Trigonometry (30°, 45°, 60°) | sin, cos, tan values | For right‑triangle ratios |
| Volume (Cube) | s³ | Simple cube problems |
| Volume (Cylinder) | πr²h | Often used in “volume of water” problems |
Quick Tips
- Always spot the shape first – this tells you which formula to use.
- Remember “½ × base × height” for triangles; it’s the only time you’ll see a factor of ½.
- Use π ≈ 22/7 for mental arithmetic if necessary.
- Check units – keep the same units throughout.
- Practice the “Pythagoras trick” – if you know two sides, you can find the third quickly.
Final Thought
Geometry in the MNS aptitude test is about recognizing patterns and applying basic formulas efficiently. Practice a few problems a day, keep this cheat‑sheet handy, and you’ll be ready to tackle any geometry question that appears on the exam. Happy studying!