We know that the polygons are those closed figures bounded by the straight line segments called as sides. The polygons are named according to the number of sides. They are also classified according to their sides and angles. You might have come across polygonal slabs fixed for the platforms or the floor of a building. These give attractive look for the surface where they are fixed. To fix certain number of polygonal tiles for the given area, we should know the area of each tile. It will be more convenient to calculate the number of tiles required for the given area, if we calculate the area of one tile. In this section let us see the methods of finding the area of different types of polygons, to enable us to solve the real life situations discussed above.





Convex Polygons: Polygons which have their interior angles acute, 90o or obtuse are called convex polygons.
Triangles, parallelograms, trapezoid, rectangle, square, kite area ll the convex polygons.
The following figures show some of the convex polygons.


Convex Polygons
Concave Polygons: Polygons which have at least one of their interior angles more than ( 180o ), reflex are called concave polygons.
The concave polygons will have its minimum number of sides four.
The following diagrams show some of the polygons which are concave.
The following figures show some of the concave polygons.


Concave Polygons
In the figure (a), the concave polygon is a quadrilateral which has one reflex angle.

In the figure ( b ), the concave polygon is a pentagon, which has one reflex angle.

In the figure ( c ), the concave polygon is a hexagon which has one reflex angle.

In the figure ( d ), the concave polygon is a hexagon with two reflex angles.
Equilateral Polygons: Polygons whose sides are congruent are called equilateral polygons.
Example: Equilateral Triangle, Square, rhombus, regular pentagon.

Equilateral Polygon

Equiangular Polygons: Polygons whose interior angles congruent are called equiangular polygons.
Example: Equiangular Triangle, Square, Rectangle.


Equiangular Triangle
Regular Polygons: Polygons whose sides are congruent and the interior angles are congruent are called regular polygons.
Example : Equilateral triangle, square, regular pentagon, regular hexagon etc.
Sum of the all the interior angle of a regular polygon = ( n - 2 ) 180o , where n is the number of sides.

Measure of each interior angle of a regular polygon =
$\frac{(n-2)180^{o}}{n}$, where n is the number of sides.

Irregular Polygons: Polygons whose sides or interior angles are unequal are called irregular polygons.
The following figure shows some of the regular and irregular polygons.
Sum of the all the interior angle of a regular polygon = ( n - 2 ) 180o , where n is the number of sides.
The measure of each interior angle will be different as the polygon is not regular.

Regular and Irregular Polygons
The following description shows the names given to the polygons according to the number of sides.

3 sides - Triangle
4 Sides - Quadrilateral
5 sides - Pentagon
6 Sides - Hexagon
7 Sides - Septagon (or) Heptagon
8 Sides - Octagon
9 Sides - Nonagon
10 Sides - Decagon
11 Sides - Hen-decagon
12 Sides - DoDecagon
13 Sides - Tri Decagon
14 Sides - Tetra Decagon
15 Sides - Penta Decagon
- - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - -
20 Sides - Icosagon
We can find the sum of interior angles of a polygon according to the number of overlapping triangles that can be formed within a polygon.

For a quadrilateral, by drawing one diagonal, we get two triangles.
since the sum of the interior angles of a triangle is 180o, the sum of the interior angles of a quadrilateral 2 times 180 = 360o

For a polygon of 5 sides, we can form three overlapping triangles.

Therefore, sum of the interior angles of a pentagon = 3 x 180o = 540o
Sum of the interior angles of a hexagon = 4 x 180o = 720o
Therefore,
for a polygon of n sides, the sum of interior angles = (n - 2) 180o

Solved Examples

Question 1: Find the sum of interior angles of a polygon of 12 sides.
Solution:
 
According to the above formula, sum of the interior angles of a polygon of n sides = (n - 2) 180o
Since number of sides = 12,
substituting n = 12, we get,
Sum of the interior angles of the polygon of 12 sides = (12 - 2) 180o
                                                                            = 10 x 180o
                                                                            = 1800o
 

Question 2: The sum of the interior angles of a polygon is 2700o , find the number of sides.
Solution:
 
We have, sum of the interior angles = 2700o
(i. e) , (n - 2) 180o = 2700
         =>          n - 2 = $\frac{2700}{180}$ = 15
         =>          n - 2 = 15
         =>               n = 15 + 2
                               = 17
Therefore, the number of sides of the polygon is 17.
Polygons
 

1. Sum of the interior angles of any polygon of n sides = (n - 2) 180o

2. Sum of each interior angle of a regular Polygon of n sides = $\frac{(n-2)180^{o}}{n}$

3. In any polygon, Interior Angle + Exterior angle = 180o

4. In a regular polygon of n sides, each exterior angle = $\frac{360^{o}}{n}$

5. The number of sides of a regular polygon, for the given exterior angle = $\frac{360^{o}}{exterior\;angle}$

Practice Problems

Question 1: The angles of a quadrilateral are, 5x, (3 x + 10)o, (6 x - 20)o¬†and (x + 25)o. Find the value of x and the measure of each angle of the quadrilateral.
Question 2: Each interior angle of a regular polygon is 144o. Find the interior angle of a regular polygon which has double the number of sides as the first polygon.
Question 3: The ratio between the interior angle and exterior angle of a polygon is 2 : 7, find the number of sides of the polygon.
Question 4: Find the number of sides of a polygon whose sum of all the interior angles is equal to 1980o.
Question 5: Find the measure of each exterior angle of a polygon of 24 sides.