Solved Examples on Area of Polygon

Example 1: Find the area of the following polygon,

Solution:

Divide the whole polygon into smaller polygons as seen in the above diagram

Clearly, the polygon obtained by joining the interior vertices of the polygon is a regular pentagon with side 7 units.

Required Area = Area of regular pentagon with side 7 units + 5(Area of equilateral triangle with side 7 units)

= [Tex]\dfrac{1}{4} × \sqrt{5(5+2√5)} × (7)^2 + 5 \times \frac{√3}4 \times 7 \times 7[/Tex]

= 683.811 sq. units

Example 2: Find the area of the following irregular polygon.

Solution:

Divide the whole polygon into smaller polygons as follows,

The figure gets divided into two rectangles- R1 and R2.

Area of polygon = Area of R₁ + Area of R₂

= 9 x 6 + 5 x 18

= 54 + 90

= 144 square meters.

Example 3: What is the perimeter and area of a regular hexagon whose side is 9 cm?

Solution:

Perimeter of the hexagon = 6 x length of side

                                         = 6 x 9

                                         = 54 cm.

Area of a regular hexagon = [Tex]\frac{3\sqrt3\times(9)^2}{2}[/Tex]

                                           = 210.45 sq. cm.

Example 4: Find the perimeter and area of a regular nonagon with a side of 10 cm.

Solution:

A regular nonagon has 9 equal sides. 

Given:

Length of each side = 10 cm.

So, Perimeter = n x l = 9 x 10 = 90 cm.

Apothem = [Tex]\frac{l}{2\tan(\frac{180}{n})}   [/Tex] = 13.73 cm

Now, Area = [Perimeter x Apothem]/2

                 = 90 x 13.73/2

                 = 617.85 sq. cm.

Example 5: Find the area of a regular polygon whose each interior angle measures 144°. Assume that each side of this polygon measures 15 units.

Solution:

Since Each Interior Angle = 144°

Each Exterior Angle = 180° – 144° = 36°

Number of sides of the polygon = 360°/Exterior Angle

                                                  = 360°/36°

                                                  = 10 sides

Thus, the given polygon is a decagon.

Apothem = [Tex]\frac{l}{2\tan(\frac{180}{n})}   [/Tex] = 23.08 units

Area = [n x l x Apothem]/2

        = [10 x 15 x 23.08]/2

        = 1731 sq. units

Example 6: Find the area of a rectangular octagon that has been cut from a square of side 8 cm.

Solution:

In order to form an octagon from a square, 4 right triangles need to be cut, each from each of the four corners of the given square. This is shown below:

Now, side of the octagon = 8(√2 – 1) = 3.31 cm

Apothem = 3.99 cm

Area = [n x l x Apothem]/2

         = 8 x 3.31 x 3.99/2

         = 52.87 sq. cm.

Area of a Polygon is the space covered inside the boundary of any polygon. Polygons are two- dimensional plane figures with at least three or more sides. It is to be noted that a polygon has a finite number of sides. The number of sides in a polygon determines its name. For example, a pentagon is a polygon that has 5 sides, a hexagon has 6 sides, a heptagon has 7 sides, and so on. Regular polygons are class