Introduction
A polygon is a shape with straight sides. “Polygon” is a Greek word that roughly translates to “many angles.” A polygon is a closed figure with three sides and three vertices at a minimum. The terms ‘poly’ and ‘gon’ mean ‘many’ and ‘angle’, respectively. As a result, polygons have a lot of angles. The perimeter and area of a polygon are determined by its type. A polygon is a two-dimensional shape enclosed by a finite number of straight lines joined in the form of a closed loop.
Parts of a Polygon
The sides or edges of a polygon are the line segments that make up the polygon. The vertex of the polygon, on the other hand, is the point where any two sides meet. Simply put, the straight lines of a polygon are called edges or sides, and its corners are called vertices. The Latin word for corner is vertex.
Interior angles and exterior angles are the two types of angles found in polygons. An exterior angle is defined as the angle through which you must turn to change your direction from one side to the next.
Polygons are classified according to the number of sides and vertices they have. An equilateral triangle, for example, is a polygon with three sides and three angles. The important polygons are those with 12 sides. The names of a polygon are a collection of words that describe the shape of a polygon. The majority of polygons have names that end in –gon. Most begin with a prefix derived from the Greek or Latin word for the number of sides.
Here is a list of the first ten polygons and their names.
Regular Polygons
Regular polygons are equiangular and equilateral. In other words, regular polygons have sides whose length, interior angles, and exterior angles are all the same.
The following are some of the most commonly used polygons in tabular form:
- The following formula can be used to calculate the sum of interior angles in a polygon with n sides:
Sum of interior angles = 180°(n – 2 )
- Since the sum of internal angles in a triangle is always 180 degrees, the sum of internal angles in a polygon is just 180 degrees multiplied by the number of triangles it can be divided into.
- We know that all the interior angles in a regular polygon are equal. The interior angles of a regular polygon with n sides can be calculated using the following formula:
Interior angle = 180°(1 – \(\frac{2}{n}\))
- The sum of all the exterior angles of a polygon is always 360°. All exterior angles are equal in a regular polygon. The exterior angle of a regular polygon with n sides can be calculated using the following formula:
Exterior angle = \(\frac{360^{\circ}}{n}\)
- The angles formed by the adjacent sides of a polygon can also be used to classify it. The angle can be less than 180 degrees or more than 180 degrees.
Convex Polygons
These polygons have interior angles that are always less than 180 degrees. A regular hexagon with six sides is referred to as a convex polygon. The vertices of a convex polygon are also protruding or pointing outwards. All the diagonals are inside the polygon.
Concave Polygons
They are the polar opposite of convex polygons, which have at least one interior angle greater than 180 degrees. The vertices of a concave polygon can also point both inwards and outwards. Diagonals may lie outside the polygon.
Solved Examples
Which of the following are concave polygons?
Solution:
A concave polygon is a polygon in which at least one of the interior angles is greater than 180°.
In a concave polygon, at least one of the diagonals is exterior to the polygon.
In the above figures,
Figures (ii) and (iii) are concave polygons since both have diagonal lying in the exterior region of the polygon.
A convex polygon is a polygon in which all the interior angles are less than 180°. In a convex polygon, all the diagonals are interior to the polygon.
Figure (i) is a convex polygon since all the interior angles are less than 180°.
A simple closed curve is called a polygon. It is made up of line segments. Polygons have been given names based on the number of sides they have. A triangle is a polygon with three sides and three angles, and a quadrilateral is a polygon with four sides and four angles.
Squares and rectangles have four sides and angles, hence they are called quadrilaterals. Since their angles are less than 180 degrees, they are convex polygons. The rectangle is an irregular polygon, whereas the square is a regular polygon.
As we can see in the figure above, the soccer ball is made of pentagons and hexagons.