Geometry Basics – Maharashtra Police Constable (Quantitative Aptitude)
Goal – Equip you with the foundational geometry concepts and formulas that frequently appear in the MP‑CPT (Police Constable) exam.
1. Simple Explanation
Geometry is the branch of mathematics that deals with shapes, sizes, and positions of objects in space.
In the exam you’ll encounter:
| Concept | What you need to know |
|---|---|
| Points, Lines, Planes | Basics – a point has no size, a line is straight and extends infinitely, a plane is a flat surface. |
| Angles | Measures between two intersecting lines. |
| Triangles | 3‑sided polygons; key types: equilateral, isosceles, scalene, right, acute, obtuse. |
| Quadrilaterals | 4‑sided polygons – rectangle, square, parallelogram, rhombus, trapezium. |
| Circles | All points equidistant from a centre. |
| Perimeter & Area | Surface measurement (perimeter) vs. area (surface inside). |
| Basic Formulas | Quick ways to compute perimeter, area, diameter, radius, etc. |
2. Key Points
2.1 Fundamental Definitions
| Term | Definition |
|---|---|
| Point | Location, no dimensions. |
| Line | Straight path extending infinitely in both directions. |
| Line Segment | Part of a line with two endpoints. |
| Ray | Line starting at a point and extending infinitely in one direction. |
| Plane | Flat, 2‑D surface extending infinitely. |
| Angle | Measure of rotation between two rays. |
| Triangle | 3 connected line segments. |
| Quadrilateral | 4 connected line segments. |
| Circle | Locus of points equidistant from a centre. |
2.2 Angles
| Angle Type | Measure |
|---|---|
| Acute | < 90° |
| Right | 90° |
| Obtuse | > 90° and < 180° |
| Straight | 180° |
| Reflex | > 180° and < 360° |
Important facts
- Sum of interior angles of a triangle = 180°.
- Sum of interior angles of a quadrilateral = 360°.
- Exterior angle of a triangle = sum of two opposite interior angles.
2.3 Triangles
| Property | Formula / Note |
|---|---|
| Perimeter | (P = a + b + c) |
| Area | (\displaystyle \frac{1}{2} \times \text{base} \times \text{height}) |
| Heron’s Formula (when all sides known) | (\displaystyle A = \sqrt{s(s-a)(s-b)(s-c)}), where (s = \frac{a+b+c}{2}) |
| Pythagoras (right triangle) | (a^2 + b^2 = c^2) (c = hypotenuse) |
2.4 Quadrilaterals
| Shape | Key Properties |
|---|---|
| Rectangle | Opposite sides equal, all angles 90°. |
| Square | All sides equal, all angles 90°. |
| Parallelogram | Opposite sides equal & parallel; opposite angles equal. |
| Rhombus | All sides equal; opposite angles equal; diagonals bisect each other at right angles. |
| Trapezium (USA: trapezoid) | At least one pair of opposite sides parallel. |
Perimeter: sum of all sides.
Area: varies by type (see below).
| Quadrilateral | Area Formula |
|---|---|
| Rectangle | (A = \text{length} \times \text{breadth}) |
| Square | (A = \text{side}^2) |
| Parallelogram | (A = \text{base} \times \text{height}) |
| Rhombus | (A = \frac{d_1 \times d_2}{2}) (product of diagonals / 2) |
| Trapezium | (A = \frac{(\text{Base}_1 + \text{Base}_2) \times \text{Height}}{2}) |
2.5 Circles
| Symbol | Meaning |
|---|---|
| (r) | Radius |
| (d) | Diameter = (2r) |
| (\pi) | Ratio of circumference to diameter (≈ 3.1416) |
| Formula | Use |
|---|---|
| Circumference | (C = 2\pi r) |
| Area | (A = \pi r^2) |
| Arc length (given central angle ( \theta) in degrees) | (L = \frac{\theta}{360^\circ} \times 2\pi r) |
2.6 Perimeter & Area – Quick Reference
| Shape | Perimeter | Area |
|---|