## what is a polygon?

In geometry, a polygon can be defined as a flat or two-dimensional closed shape bounded by straight sides. it has no 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 some examples of polygons.**

**here are some examples of a polygon**

## polygon graph

Polygons are named by the number of sides they have. polygons are usually denoted by n-gon where n represents the number of sides it has, for example, a five-sided polygon is called a 5-gon, a ten-sided one is called a 10-gon, and so on.

However, few polygons have 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.

Although polygons with sides greater than 10 also have special names, we usually denote them with n-gon since the names are complex and not easy to remember.

## polygon types

Polygons can be classified based on the number of sides and angles they have:

**classification according to sides: regular and irregular polygons:****classification based on angles: convex and concave polygons:**- sides are equal
- interior angles are equal
- exterior angles are equal
- the name of the three-sided regular polygon is ________________.
- A regular polygon is a polygon whose _____________ are equal and all angles are equal.
- The sum of the exterior angles of a polygon is __________.
- A polygon is a simple closed figure made up of only _______________.
- equilateral triangle
- sides
- 360°
- line segments
**nonagon****triangle****pentagon****decagon**- 9
- 3
- 5
- 10

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

For example, an equilateral triangle is a regular polygon with three sides. A square is a regular polygon with four sides. a regular hexagon is a regular polygon with six sides.

here are some examples of regular polygons.

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

here are some examples of irregular polygons.

**convex polygons: **A convex polygon is a polygon with all interior angles less than 180°.

in convex polygons, all diagonals are inside the polygon.

(diagonal is a line segment joining two non-consecutive vertices of a polygon)

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here are some 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 inside the polygon.

here are some 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 intersect one or more times.

**sum of the 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 6-sided polygon

here, ∠a + ∠b + ∠c + ∠d + ∠e + ∠f = (6 – 2) × 180° = 720° (n = 6 since the polygon has 6 sides)

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

sum of the exterior angles of the polygons = 360°

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the sum will always equal 360 degrees, regardless of how many sides it has.

for example: consider the following 5-sided polygon

here, ∠m + ∠n + ∠o + ∠p + ∠q = 360°

**angles in regular polygons**

in a regular polygon, all its

**interior angle: **

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

so, each interior angle = (n – 2) × 180n

**exterior angle:**

sum of the exterior angles of the 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 the sum of one interior angle and one exterior angle is 180°.

## solved examples on polygon

**example 1: fill in the blank.**

**solution:**

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

**solution:**

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

**solution:**

the polygon has 20 sides. so, n = 20.

sum of the exterior angles of the 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.