# Polygon – Shape, Types, Formulas, Examples

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## 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

## 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:

1. 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.

1. 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.

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

Solution:

1. equilateral triangle
2. sides
3. 360°
4. line segments

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

1. Nonagon
2. Triangle
3. Pentagon
4. Decagon

Solution:

1. 9
2. 3
3. 5
4. 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

1

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

360°
540°
720°
900°
CorrectIncorrect
Heptagon has 7 sides. So, n = 7, Sum of the angles of a polygon = (n – 2) × 180° = (7 – 2) × 180° = 5 × 180° = 900°
2

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

180°
360°
540°
720°
CorrectIncorrect
The sum of exterior angles of a polygon will always be equal to 360 degrees, irrespective of the number of sides it has.
3

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

Regular polygon
Concave polygon
Convex polygon
CorrectIncorrect
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°.
4

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

Square
Equilateral triangle
Rhombus
Hexagon with equal sides
CorrectIncorrect