Geometry Basics – Quick Study Notes
(Suitable for the Jammu & Kashmir Police Constable Quantitative Aptitude exam)
1. Simple Explanation
Geometry deals with the properties of points, lines, angles, surfaces and solids.
For the exam you need to know the basic formulas and be able to apply them quickly to solve questions on:
| Shape | Key Feature | What to Remember |
|---|---|---|
| Line | Straight, one‑dimensional | Length only |
| Ray | Starts at a point and extends infinitely | One endpoint, infinite length |
| Segment | Finite part of a line | Two endpoints, finite length |
| Angle | Formed by two rays with a common vertex | Measured in degrees (°) |
| Triangle | Three sides, three angles | Types: equilateral, isosceles, scalene, right, obtuse, acute |
| Quadrilateral | Four sides | Rectangle, square, rhombus, parallelogram, trapezium |
| Circle | All points equidistant from centre | Radius (r), diameter (2r), circumference, area |
| Polygon | Closed shape with straight sides | Regular (all sides/angles equal) vs irregular |
2. Key Points & Formulas
| Concept | Formula | Variables | Notes |
|---|---|---|---|
| Perimeter | (P = 2\pi r + 2\pi r) (circle) | (r) = radius | For polygons sum of side lengths |
| Circumference | (C = 2\pi r) | (r) | Use (\pi = 3.14) unless specified |
| Area (Circle) | (A = \pi r^2) | (r) | |
| Area (Rectangle) | (A = l \times w) | (l) = length, (w) = width | |
| Area (Parallelogram) | (A = b \times h) | (b) = base, (h) = height | |
| Area (Trapezium) | (A = \dfrac{(a+b)}{2}\times h) | (a,b) = parallel sides, (h) = height | |
| Area (Triangle) | (A = \dfrac{1}{2} \times b \times h) | (b) = base, (h) = height | Also (A = \sqrt{s(s-a)(s-b)(s-c)}) (Heron) |
| Pythagoras | (a^2 + b^2 = c^2) | (c) = hypotenuse | Only for right‑angled triangles |
| Angle Sum | (\text{Sum of interior angles of } n\text{-gon} = (n-2)\times 180^\circ) | ||
| Right‑Angle Properties | In a right triangle, the side opposite the right angle is the hypotenuse | ||
| Symmetry | Reflection, rotational, translational symmetry |
3. Worked Example
Problem
A rectangle has a length of 12 cm and a width 5 cm.
- Find its perimeter and area.
- A circle is inscribed inside this rectangle such that the circle touches all four sides. Find the radius, circumference and area of the circle.
Solution
| Step | Calculation | Result |
|---|---|---|
| 1. Perimeter of rectangle | (P = 2(l + w) = 2(12 + 5) = 34) cm | (P = 34) cm |
| 1. Area of rectangle | (A = l \times w = 12 \times 5 = 60) cm² | (A = 60) cm² |
| 2. Radius of inscribed circle | Since the circle touches all sides, its diameter equals the smaller side of the rectangle → (d = 5) cm → (r = d/2 = 2.5) cm | (r = 2.5) cm |
| 2. Circumference | (C = 2\pi r = 2 \times 3.14 \times 2.5 = 15.7) cm | (C \approx 15.7) cm |
| 2. Area | (A = \pi r^2 = 3.14 \times (2.5)^2 = 3.14 \times 6.25 = 19.625) cm² | (A \approx 19.63) cm² |
Answer
- Rectangle: perimeter = 34 cm, area = 60 cm²
- Inscribed circle: radius = 2.5 cm, circumference ≈ 15.7 cm, area ≈ 19.63 cm²
4. Quick Revision Summary
| Topic | Formula / Concept | Quick Tip |
|---|---|---|
| Perimeter | Sum of side lengths | Easy: just add |
| Circumference | (2\pi r) | Remember π ≈ 3.14 |
| Area of Circle | (\pi r^2) | Use (r^2) first, then multiply |
| Area of Rectangle/Parallelogram | (l \times w) or (b \times h) | Base × height |
| Area of Trapezium | (\frac{(a+b)}{2} \times h) | Average of parallel sides × height |
| Area of Triangle | (\frac{1}{2} \times b \times h) | Half base × height |
| Pythagoras | (a^2 + b^2 = c^2\ |