Sales Toll Free No: 1-855-666-7446

# Polygons

Top
 Sub Topics The geometry is a special subject in which we deal with different types of shapes known as geometrical figures or objects. A geometrical figure is a predefined shape that satisfies certain properties specified for that particular shape. The polygons are among the most important geometrical shapes. The polygons actually define a family of geometrical figures. They are defined as the closed figures that are formed by three or more line segments. Recall that a line segment is a portion of a line. It has a fixed length and two fixed endpoints.The term "polygon" is a combination of two Greek language words - poly and gonia. The former meant "many", while the latter indicates "angle" or "corner". Thus, a polygon is said to be a plane figure having many angles. This shape is bounded with the finite number of straight line segments, together forming a closed circuit. Such line segments are known as sides or edges of the polygon. The point of joining of any two sides forms a corner which is called a vertex of the polygon. The number of sides in a polygon is termed as its dimension, for instance - a polygon having n sides is said to be an n-gon. Let us go ahead and learn more about the polygons, their different types and their properties in this article below.

## Types

There are many different types of polygons. They are mainly classified into two on the basis of equal and unequal sides, as written below:

1) Regular Polygon:  If a polygon has all sides equal, then we say that the polygon is a regular polygon.

2) Irregular Polygon: On the other hand, if a polygon has unequal sides, then it is to be said as an irregular polygon.

Both regular as well as irregular polygons can be classified according to the number of sides they contain. This classification is discussed below:

1) Triangle: A polygon with three line segments. It has three sides, three vertices and three angles.

2) Quadrilateral: The quadrilaterals are 4-sided polygons. They include a number of polygonal shapes, such as square, rectangle, kite, trapezium, rhombus and parallelogram.

3) Pentagon: It is a five-sided polygon.

4) Hexagon: A hexagon is said to a six-sided polygon.

5) Heptagon or Septagon: A polygon with seven sides is known as a heptagon or septagon.

6) Octagon: An eight-sided polygon is called an octagon,

7) Nonagon: A polygon having nine edges is a nonagon.

8) Decagon: A ten-sided polygon is termed as decagon.

9) Hendecagon: It is an eleven-sided polygon.

10) Dodecagon: Dodecagon is a polygon with twelve sides.

and so on.
We may have a look at the diagrams of different types of polygons.

## Properties

The properties of polygons are listed below:

i) The sum of all the interior angles of a polygon can by calculated by the following formula:

Sum of interior angles = (n - 2) x 180$^{\circ}$

where, n represents the number of sides in the polygon.

ii) A diagonal in a polygon is defined as a line segment joining any of its two vertices. In a polygon, there exists more than one diagonals. The number of diagonals in a polygon are determined by the formula = $\frac{1}{2}$ n(n - 3).

iii) When we draw all the diagonals from one particular vertex of a polygon, the triangles are formed. We may also find how many triangles can be formed by using the formula, number of triangles = (n - 2).

iv) A polygon having all the interior angles less than 180$^{\circ}$ is called convex polygon. On the other hand, if a polygon has at least one angle as reflex angle, then it is said to be concave triangle. It is shown in the figure below :

If a line drawn through a convex polygon, then it intersects it in two points. But when we draw a line through a concave polygon, then it intersect it in more than two points.

## Area

The formula for the area of a regular polygon is given below:

Area = $\frac{1}{2}$ perimeter x apothem

Where, apothem of a regular polygon is defined as the line segment joining the center of polygon and the midpoint of any side.

Let perimeter is denoted by p and apothem by a, then area A is given by :

A = $\frac{1}{2}$ p a

If the length of each side in a regular polygon is "s" and there be n number of sides, then the perimeter would be equal to n x s. Hence, the formula of area becomes:

A = $\frac{1}{2}$ n s a

The value of apothem a can be calculated by:

a = $s\ cot$ ($\frac{\pi}{n}$)

Therefore, the area of a regular polygon can also be written as:

A = $\frac{1}{2}$ $n\ s^{2}\ cot$($\frac{\pi}{n}$)

## Angles

A regular polygon has a property that it has all the interior angles equal. Thus, we can easily determine the value of interior angle in a regular polygon if the number of sides are known. The formula for measure of each angle is given by :
Angle of Regular Polygon = $(n - 2)$ . $\frac{180^{\circ}}{n}$
Where, n denotes the number of sides in the polygon.

We may also find the sum of all the interior angles if number of sides are given, by using the following relation:

Sum of Interior Angles = $(n - 2). 180^{\circ}$

## Perimeter

P = n $\times$ s