Polygon Formula: Definition, Types & Properties

A polygon is a 2-D shape that contains three or more straight lines. The word Polygon is derived from the Greek language in which the word “Poly” signifies “many”, and the word “Gon” means “angle”. So, it can be said that a closed figure which has many angles is a Polygon. 

• Some of the major examples of polygons are Triangles, squares, rectangles, pentagons, and hexagons. 
• A point at which the segments of a polygon meet is referred to as the vertices. 
• Similarly, the segments that share the same vertex are referred to as the adjacent sides. 
• A segment whose endpoints are nonadjacent vertices is also known as a diagonal. 

Key Terms: Polygon, Triangle, Square, Rectangle, Hexagon, Interior Angles

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What is a Polygon?

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Polygons must have at least three sides. Each side of the line segment should intersect with the other line segment at a point which is called a vertex.

• Simply, a point at which segments meet is referred to as the vertices. 
• Segments that share the same vertex are referred to as the adjacent sides. 
• A segment whose endpoints are nonadjacent vertices is called a diagonal. 
• Triangles, squares, rectangles, pentagons, and hexagons, are some examples of polygons.
• There is also a special class of polygon whose sides and angles are of the same length, such polygons are called regular polygons. No matter how one lays that polygon down, it will look the same from all four sides.

Polygon Diagram

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Polygon Formulas

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Sum total of all internal angles: T =( n - 2) X 180

Where,

• T = Sum of internal angles
• n = Number of sides

Each Interior Angle: IA = [(n−2)180] / n 

Where, 

• IA = Each internal angle
• n = Number of sides

Each Exterior Angle: EA = 360/n

Where,

• EA = Each exterior angle
• n = Number of sides

Perimeter of a Polygon: P = n × s

Where,

• P = Perimeter
• n = Number of sides
• s = The length of the sides

Area of the polygon: A= s/2 tan (180/n) 

Where,

• A = Area of Polygon
• s = the length of the sides
• n = Number of sides

The number of diagonals: D= [n(n−3)]/ 2

Where, 

• D = Diagonal of Polygon
• n = Number of sides

Polygon Types

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Types of Polygons

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Based on the sides and the angle measure the polyones can be classified into different types. These are:

Regular Polygon- All the sides as well as interior angles are of the same measure. Given below is an example of a Regular Polygon.

Regular Polygon

Irregular Polygon- All the sides as well as angles are of different measures in an irregular polygon. Given below is an example of an Irregular Polygon.

Irregular Polygon

Convex Polygon- All the interior angles of a convex polygon are less than 180 degrees. Given below is an example of a Convex Polygon.

Convex Polygon

Concave Polygon- These are the Polygons that have one or more interior angles with a measure of greater than 180 degrees. Given below is an example of a Concave Polygon.

Concave Polygon

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List of Polygons

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Generally the number of sides of a polygon determines its shape and on the basis of sides, a polygon is named. Common examples are triangles, squares, pentagons, hexagons, etc. Mentioned below is the list of polygons based on the number of sides they have:

List of Polygons

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Properties of Polygons

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Important properties of Polygons are:

• The sum of all the interior angles of all the quadrangles is 360°.
• If at least one of the interior angles is greater than 180 degrees, then it is called a concave polygon.
• If a polygon does not cross over itself and has only one boundary, it is known as a simple polygon; otherwise, it is referred to as a complex polygon.
• An angle inside the polygon at one of its vertices is called an interior angle. 
• An angle outside the polygon formed by one of its sides and the extension of an adjacent side is called the exterior angle of the quadrilateral.
• The Interior angle plus the corresponding exterior angle is equal to 1800.

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Things to Remember

• A polygon is a 2-D shape that contains three or more straight lines. 
• Sum total of all internal angles of a Polygon is T =( n – 2) X 180( Here, T= Sum of internal angles and n= Number of sides )
• Each Interior Angle of a Polygon is IA = [(n−2)180] / n (Here, IA= Each internal angle and n= Number of sides)
• Each Exterior Angle of a Polygon is EA = 360/n (Here, EA= Each exterior angle and n= Number of sides)
• Perimeter of a Polygon is P=n×s (Here, P= Perimeter, n= Number of sides and s= The length of the sides)
• The Area of the polygon is A= s/ 2 tan (180/n) (Here, A= Area of Polygon, s= the length of the sides and n= Number of sides)
• The number of diagonals in a Polygon is D= [n(n−3)]/ 2 (Here, D= Diagonal of Polygon and n= Number of sides)