## What is a Polygon?

In geometry, a **polygon can be defined as a flat or plane, two-dimensional closed shape bounded with straight sides**. It does not have curved sides. **The sides of a polygon are also called its edges**. The points where two sides meet are the vertices (or corners) of a polygon.

**Here are a few examples of polygons.**

**Here are a few non-examples of a polygon**

#### Begin here

Polygon Chart

Polygons are named on the basis of the number of sides it has. **Polygons are generally denoted by n-gon where n represents the number of sides it has**, For example, a five-sided polygon is named as 5-gon, a ten-sided is named as 10-gon, and so on.

However, few polygons have some special names. The minimum number of sides a polygon can have is 3 because it needs a minimum of 3 sides to be a closed shape or else it will be open.

Even though polygons with sides greater than 10, also have special names, we generally denote them with n-gon as the names are complex and not easy to remember.

## Types of Polygon

The polygons can be classified on the basis of the number of sides and angles it has:

**Classification on the basis of sides: Regular and Irregular Polygons:**

**Regular Polygons** – Polygons that have equal sides and angles are regular polygons.

For example, an equilateral triangle is a three-sided regular polygon. A square is a four-sided regular polygon. A Regular hexagon is a six-sided regular polygon.

Here are a few examples of regular polygons.

**Irregular Polygons** – Polygons with unequal sides and angles are irregular polygons.

Here are a few examples of irregular polygons.

**Classification on the basis of angles: Convex and Concave Polygons:**

**Convex Polygons – **A convex polygon is a polygon with all interior angles less than 180°.

In convex polygons, all diagonals are in the interior of the polygon.

(*Diagonal is a line segment joining any two non-consecutive vertices of a polygon*)

Here are a few examples of convex polygons.

**Concave Polygons – **A concave polygon is a polygon with at least one interior angle greater than 180°.

In concave polygons, not all diagonals are in the interior of the polygon.

Here are a few examples of concave polygons.

**Difference between Convex and Concave Polygon**

**3. Simple and Complex Polygon:**

**Simple Polygon –** A simple polygon has only one boundary. The sides of a simple polygon do not intersect.

**Complex Polygon – **Complex polygon is a polygon whose sides cross over each other one or more times.

**Sum of Angles of a Polygon**

**1. Sum of the interior angles of a polygon:**

Sum of the interior angles of a polygon with n sides = (n – 2) × 180°

For example: Consider the following polygon with 6 sides

Here, ∠*a* + ∠*b* + ∠*c* + ∠*d* + ∠*e* + ∠*f* = (6 – 2) × 180° = 720° (n = 6 as given polygon has 6 sides)

**2. Sum of the exterior angles of polygons**

Sum of the exterior angles of polygons = 360°

The sum will always be equal to 360 degrees, irrespective of the number of sides it has.

For example: Consider the following polygon with 5 sides

Here, ∠*m* + ∠*n* + ∠*o* + ∠*p* + ∠*q* = 360°

**Angles in Regular Polygon**

In a regular polygon, all its

- sides are equal
- interior angles are equal
- exterior angles are equal

**Interior Angle: **

Sum of the interior angles of a polygon with n sides = (n – 2) × 180°

So, each interior angles = (n – 2) × 180n

**Exterior Angle:**

Sum of the exterior angles of polygons = 360°

So, each exterior angle = 360°n

**Sum of Interior Angle and Exterior Angle:**

Whether the polygon is regular or irregular, at each vertex of the polygon sum of an interior angle and exterior angle is 180°.

## Solved Examples on Polygon

**Example 1: Fill in the blank.**

- The name of the three sided regular polygon is ________________.
- A regular polygon is a polygon whose all _____________ are equal and all angles are equal.
- The sum of the exterior angles of a polygon is __________.
- A polygon is a simple closed figure formed by only _______________.

**Solution:**

- equilateral triangle
- sides
- 360°
- line segments

**Example 2: Write the number of sides for a given polygon.**

**Nonagon****Triangle****Pentagon****Decagon**

**Solution:**

- 9
- 3
- 5
- 10

**Example 3: Find the measure of each exterior angle of a regular polygon of 20 sides.**

**Solution:**

The polygon has 20 sides. So, n = 20.

Sum of the exterior angles of polygons = 360°

So, each exterior angle = 360°n = 360°20 = 18°

**Example 4: The sum of the interior angles of a polygon is 1620°. How many sides does it have?**

**Solution:**

Sum of the interior angles of a polygon with n sides = (n – 2) × 180°

1620° = (n – 2) × 180°

n – 2 = 1620180

n – 2 = 9

n = 9 + 2

n = 11

So, the given polygon has 11 sides.

## Practice Problems on Polygon

## Polygon

### What is the sum of all the angles of a heptagon?

Heptagon has 7 sides. So, n = 7, Sum of the angles of a polygon = (n – 2) × 180° = (7 – 2) × 180° = 5 × 180° = 900°

### What is the sum of the exterior angle of a pentagon?

The sum of exterior angles of a polygon will always be equal to 360 degrees, irrespective of the number of sides it has.

### If three angles of a quadrilateral are each equal to 55°, the quadrilateral is of type:

Quadrilateral has 4 sides. So, n = 4 Sum of the angles of a polygon = (n – 2) × 180° = (4 – 2) × 180° = 2 × 180° = 360° 55° + 55° + 55° + fourth angle = 360° Measure of fourth angle = 360° - 165° = 195° So, the given quadrilateral is concave polygon as it has at least one interior angle greater than 180°.

### Rhombus has all sides equal but it’s all angles are not equal.

Which of the following is not a regular polygon?

**F**requently Asked Questions on Polygon

**What is the diagonal of a polygon?**

A diagonal of a polygon is a line segment connecting two non-consecutive vertices (corners).

**What are the properties of regular polygons?**

A Regular polygon has all sides of equal length and each angle also measures equal.

**Is a circle a polygon?**

Polygon is a closed shape made up of straight-line segments. The circle is a closed figure but it is made of a curve. So, a circle is not a polygon.

**What is the minimum number of sides a polygon must-have?**

A polygon must have a minimum of three sides.

**Can a number of angles and the number of sides for a polygon be different?**

No, polygons have the same number of sides and angles because they are closed figures with non-intersecting lines.