CAT 2025 Lesson : Quadrilaterals - Properties of Polygons

1. Introduction

In the management entrance tests, questions from quadrilaterals are conceptually more advanced than those pertaining to polygons of 5 or more finite sides. Accordingly this lesson covers concepts pertaining to quadrilaterals in detail and the basic properties of other polygons with 5 or more finite sides.

A polygon is a shape enclosed by 3 or more intersecting lines. A triangle is a polygon with 3 sides. We covered triangles in the previous lesson. We will start this lesson with the basic properties that apply to all polygons, the concept of regular polygons and then learn about quadrilaterals.

2. Basics of Polygons

An n-sided polygon will be bound by $n$ intersecting lines with $n$ vertices, $n$ edges and $n$ interior (and $n$ exterior) angles. In the 5-sided polygon (pentagon) shown below, the

(a) 5 vertices are A, B, C, D and E (b) 5 edges are the line segments AB, BC, CD, DE and EA. (c) 5 interior angles are $\angle$1, $\angle$2, $\angle$3, $\angle$4 and $\angle$5. (d) 5 exterior angles are $\angle$6, $\angle$7, $\angle$8, $\angle$9 and $\angle$10.

2.1 Properties of Polygons

In an n-sided polygon (where $n$ > 3), from any vertex, we can draw diagonals to all vertices other than itself and its two adjacent vertices. So, from 1 vertex we can draw $(n - 3)$ diagonals.

$(n - 3)$ cuts will divide a polygon into $(n - 2)$ triangles. So, sum of interior angles is the sum of angles of the $(n - 2)$ triangles formed.

∴ Property 1: Sum of interior angles of an $n$-sided polygon $= (n - 2) \times 180^{\mathrm{o}}$



If each side is extended in 1-direction across vertices, exterior angles are formed. At each vertex, we have a linear pair (sum of angles is $180^{\mathrm{o}}$) comprising of 1 interior angle and 1 exterior angle.

Sum of exterior angles $=$ Sum of linear pairs - Sum of interior angles

$= n \times 180^{\mathrm{o}} - (n - 2) \times 180^{\mathrm{o}}$

= $180^{\mathrm{o}} (n - n + 2)$

= $360^{\mathrm{o}}$

∴ Property 2: Sum of exterior angles of an $n$-sided polygon = $360^{\mathrm{o}}$



Diagonals are lines that connect any 2 vertices of the polygon, that are not adjacent to each other.
(Note: Triangle is the only polygon with 0 diagonals, as all vertices are adjacent to each other.)

As explained at the start of this page, from any 1 vertex we can draw ($n$ - 3) diagonals.

From $n$ vertices we can draw $n(n - 3)$ diagonals. But, we are counting each diagonal twice. (e.g., AC and CA are counted as 2 different diagonals).

∴ Property 3: Number of diagonals in an n-sided polygon = $\dfrac { n (n - 3)}{2}$