Polygons can be found everywhere in our surroundings as well as in geometrical math. Many objects are in the shapes of polygons. In this article, students will learn what are polygons as well as various polygon formula. Some examples will help to understand the concept and formula. Let us begin the concept!

**Polygon Formula**

**What is Polygon?**

A polygon is any two-dimensional or 2D shape formed with the straight lines. Triangles, quadrilaterals, pentagons, and hexagons are related shapes. The name tells us that how many sides the shape has. For example, a triangle is having three sides, and a quadrilateral has four sides.

Therefore, any shape that can be drawn by connecting three straight lines is a triangle, and any shape that can be drawn by connecting four straight lines is called a quadrilateral. All of these shapes are polygons. If the shape had curves it will not be a polygon.

A special class of polygon exists in mathematics, it happens for polygons whose sides are all the same length and whose angles are all the same. This is called regular polygons. A stop sign as the traffic signal is an example of a regular polygon with eight sides.

All the sides are the same and no matter how you lay it down, it will look the same. You wouldn’t be able to tell which way was up because all the sides are the same and all the angles are the same.

A pentagon with all sides of equal length and equal angles, then it is called a regular pentagon.

Polygon is a word taken from the Greek language, where poly means many and gonna means angles. So we can say that in a plane, a closed figure with many angles is called a polygon. There are many properties in a polygon such as sides, diagonals, area, angles, etc.

**Some popular Polygons are:**

- Triangle – 3 sides
- Quadrilateral – 4 sides
- Pentagon – 5 sides
- Hexagon – 6 sides
- Heptagon – 7 sides
- Octagon – 8 sides
- Nonagon – 9 sides
- Decagon – 10 sides

**Various Polygon Formula**

**Sum total of all internal angles:**

\( ( T = (n-2)\times 180 \)

Where,

T | Sum of internal angles |

n | Number of sides |

**Each Interior Angle:**

IA = \(\frac{(n-2)\times 180}{n}\)

Where,

IA | Each internal angle |

N | Number of sides |

**Each Exterior Angle:**

EA = \(\frac{360}{n}\)

Where,

EA | Each exterior angle |

N | Number of sides |

**Perimeter:**

\(P = n \times s \)

Where,

P | Perimeter |

n | Number of sides |

**Area of the polygon:**

\(A = \frac{s}{2tan(\frac{180}{n})} \)

Where,

A | Area of polygon |

n | Number of sides |

**Solved Examples**

Q.1: A polygon is an octagon and its side length is 5 cm. Calculate its perimeter and value of one interior angle.

Solution:

Given in the problem:

The polygon is an octagon. Hence, n = 8

Length of one side,

s = 5 cm

The perimeter of the octagon

P = n × s

P = 8× 5

=40 cm.

Now, to compute interior angle,

IA = \(\frac{(n-2)\times 180}{n}\)

= \( \frac{(8-2)\times 180}{8}\)

= \(\frac {1080}{8}\)

\(IA = 135 \;degrees \)

So, each interior angle will be \(135 \; degrees \)

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