Geometry Basics – Study Notes
(For Nagaland Patwari / Lekhpal – Quantitative Aptitude)
Geometry is the branch of mathematics that deals with the properties and measurements of points, lines, angles, surfaces, and solids. In the context of land records and revenue work, the most useful parts are the basic shapes, their perimeter and area, and the relationship between linear and planar measurements.
1. Simple Explanation
| Term | What it Means | Quick Picture |
|---|---|---|
| Point | A location with no size. | • |
| Line | A set of points extending infinitely in both directions. | ─ |
| Line segment | Part of a line bounded by two endpoints. | ─ |
| Ray | A line with one fixed endpoint that extends infinitely in one direction. | ► |
| Angle | The figure formed by two rays sharing a common endpoint (vertex). | ∠ |
| Parallel lines | Lines in the same plane that never meet. | |
| Perpendicular lines | Lines that meet at a right angle (90°). | ⊥ |
| Triangle | A closed shape with 3 sides. | △ |
| Quadrilateral | A closed shape with 4 sides. | □ |
| Circle | All points equidistant from a center. | ○ |
| Perimeter | Total length of the boundary. | P = sum of side lengths |
| Area | Measure of the surface inside a shape. | A = length × breadth (rectangles) etc. |
| Unit of area | Commonly square feet (sq ft), square meters (sq m), or acres (1 acre = 43,560 sq ft). |
Why it matters:
- Perimeter helps in fencing calculations.
- Area is needed for tax, rent, and subdivision.
- Knowing units ensures accurate record‑keeping.
2. Key Points
| Concept | Formula / Rule | Notes |
|---|---|---|
| Rectangle / Square | Perimeter = 2 × (l + b) Area = l × b |
For a square, l = b. |
| Triangle | Perimeter = a + b + c Area (base × height ÷ 2) |
Height is the perpendicular distance from base to opposite vertex. |
| Parallelogram | Perimeter = 2 × (b + s) Area = base × height |
s = side length. |
| Rhombus | Perimeter = 4 × side Area = (d₁ × d₂) ÷ 2 (product of diagonals ÷ 2) |
Diagonals are perpendicular bisectors. |
| Trapezium (UK) / Trapezoid (US) | Perimeter = sum of all four sides Area = (a + b) × h ÷ 2 (a & b are parallel bases) |
h = height (distance between parallel sides). |
| Circle | Circumference = 2πr Area = πr² |
π ≈ 3.1416. |
| Conversion | 1 acre = 43,560 sq ft | Useful for large plots. |
| Coordinate Geometry (if needed) | Distance between (x₁,y₁) & (x₂,y₂) = √[(x₂–x₁)² + (y₂–y₁)²] | Helps in mapping irregular plots. |
3. Worked Example
Problem
A land plot consists of a rectangular block and an adjoining triangular extension.
- Rectangle: length = 120 ft, breadth = 80 ft.
- Triangle: base = 80 ft (same as rectangle’s breadth), height = 30 ft.
Find:
a) Total perimeter of the combined plot.
b) Total area of the combined plot.
c) Express the area in acres (rounded to 2 decimal places).
Solution
a) Perimeter
- Rectangle perimeter
[ P_{\text{rect}} = 2(l+b) = 2(120+80) = 2(200) = 400 \text{ ft} ] - Triangle perimeter
- Two equal sides (isosceles) are found using Pythagoras:
[ \text{Side} = \sqrt{b^2 + h^2} = \sqrt{80^2 + 30^2} = \sqrt{6400+900}= \sqrt{7300}\approx 85.44 \text{ ft} ] - Triangle perimeter
[ P_{\text{tri}} = b + 2 \times \text{Side} = 80 + 2(85.44) \approx 80 + 170.88 = 250.88 \text{ ft} ]
- Two equal sides (isosceles) are found using Pythagoras:
- Combined perimeter
The rectangle’s right side (80 ft) is shared with the triangle’s base, so we subtract it once:
[ P_{\text{total}} = P_{\text{rect}} + P_{\text{tri}} - 80 = 400 + 250.