Geometry Basics – Odisha Patwari / Lekhpal
(Quantitative Aptitude – Chapter 1)
Objective – Understand the most common geometric shapes, their properties, and how to compute length, area, perimeter, and volume that appear in exam questions.
1. Simple Explanation
| Concept | What it is | Why it matters for the exam |
|---|---|---|
| Point | Zero‑dimensional location | Used only to refer to positions. |
| Line | Straight set of points extending infinitely in both directions | Helps in defining angles, segments, and shapes. |
| Line Segment | Part of a line with two endpoints | Length is always positive. |
| Angle | Formed by two rays sharing a vertex | Measured in degrees – right(90°), acute(<90°), obtuse(>90°), straight(180°). |
| Polygon | Closed figure with straight sides | Common shapes: triangle, quadrilateral, pentagon, etc. |
| Circle | Set of points equidistant from a centre | Key for area, circumference, sector etc. |
| Area | Measure of surface inside a shape | Needed for revenue assessment, taxation, etc. |
| Perimeter / Circumference | Sum of side lengths | Often used to calculate fencing, boundary lengths. |
| Volume | Measure of space inside a 3‑D shape | Less common but useful for storage calculations. |
2. Key Points & Formulas
2.1 Basic Geometry Units
| Quantity | Symbol | Formula | Notes |
|---|---|---|---|
| Perimeter of a rectangle | (P) | (2(l + w)) | (l) = length, (w) = width |
| Area of a rectangle | (A) | (l \times w) | |
| Perimeter of a square | (P) | (4a) | (a) = side |
| Area of a square | (A) | (a^2) | |
| Perimeter of a triangle (any) | (P) | (a + b + c) | sides (a,b,c) |
| Area of a triangle | (A) | (\frac{1}{2}\times \text{base}\times \text{height}) | |
| Area of a circle | (A) | (\pi r^2) | (r) = radius |
| Circumference of a circle | (C) | (2\pi r) | |
| Area of a trapezium (trapezoid) | (A) | (\frac{1}{2}(a+b)h) | (a,b) = parallel sides, (h) = height |
| Area of a parallelogram | (A) | (\text{base} \times \text{height}) | |
| Area of a right‑angled triangle | (A) | (\frac{1}{2}ab) | (a,b) are the two legs |
| Area of a circle sector | (A) | (\frac{\theta}{360}\pi r^2) | (\theta) in degrees |
2.2 Angle Relationships
| Relationship | Condition | Example |
|---|---|---|
| Sum of angles in a triangle | (180^\circ) | If two angles are (30^\circ) and (50^\circ), the third is (100^\circ). |
| Sum of angles in a quadrilateral | (360^\circ) | |
| Supplementary angles | Add to (180^\circ) | Two angles (60^\circ) and (120^\circ) |
| Complementary angles | Add to (90^\circ) | Two angles (45^\circ) and (45^\circ) |
| Vertical angles | Equal | Angles formed by two intersecting lines |
2.3 Circle Properties
| Property | Formula | Used for |
|---|---|---|
| Chord length | (2r\sin\frac{C}{2}) | Finding chord from central angle |
| Arc length | (\frac{\theta}{360}\times 2\pi r) | Length of an arc |
| Area of segment | (A_{\text{segment}} = \frac{r^2}{2}(\theta - \sin\theta)) | Sub‑area of circle cut by chord |
Tip: For quick calculations, use (\pi \approx 22/7) or 3.14.
3. Worked Example
Problem (Typical Patwari Question)
A rectangular plot of land has a length of 120 m and a width of 80 m.
The plot is fenced with a single boundary.
How much fencing material (in metres) is needed?
Also, if the plot is divided into 4 equal rectangular sub‑plots, what is the total length of fencing required inside the plot (including the outer boundary)?
Step‑by‑Step Solution
-
Outer perimeter of the main rectangle
[ P_{\text{outer}} = 2(l + w) = 2(120 + 80) = 2(200) = 400 \text{ m} ] -
Dividing into 4 equal rectangles
- Each sub‑plot will have half the length and half the width of the main plot:
[ l_{\text{sub}} = \frac{120}{2} = 60 \text{ m},\quad w_{\text{sub}} = \frac{80}{2} = 40 \text{ m} ] - There are 3 vertical fences (running along width) and 3 horizontal fences (running along length) inside the plot.
- Length of each vertical fence = width of main plot = 80 m.
Total vertical inside fences: (3 \times 80 = 240) m. - Length of each horizontal fence = length of main plot = 120
- Each sub‑plot will have half the length and half the width of the main plot: