Area of a Pentagon Formula

To find the area of a pentagon, we use the following area of pentagon equation, 

\[A=\frac{1}{4}\sqrt{5(5+2\sqrt{5})}s^{2}\]

Note that this formula is used to find the area for any pentagon, whether it is regular or irregular.

The area of a regular pentagon can be calculated in 2 ways. 

Firstly, the area of a pentagon is given by 5/2 x s x a; in which ‘s’ represents the side of the Pentagon, and ‘a’ represents the apothem length. Apothem is the line from the centermost point of the pentagon to a side, bisecting the side at 90 degrees right angle.

How to Calculate the Area of a Pentagon?

Find the area of the pentagon given below

[Image will be uploaded soon]     

Solution:

To find out the area of the given pentagon, we would divide the interior of the pentagon into a 4-sided rectangle and two right triangles. The area of the bottom rectangle can be calculated using the formula:

 A = width × length

A = 20 × 15

    = 300

The area of the two right triangles can be calculated using the formula: 

A = base × height / 2

A = 10 × 8/2

  = 80/2

  = 40

Seeing that there are two right triangles, the sum of both will be equivalent to the area of the entire triangular top portion of the pentagon.

Hence, the solution is:

A = 300 + 40 + 40

   = 380

Perimeter of Pentagon Formula

In a regular pentagon, all of the sides measure equal in length. The perimeter of a pentagon is actually the total length of its boundaries. Thus, it is equivalent to the sum of all its five sides. For a regular pentagon,

The perimeter can then be calculated by summing up all 5 sides or multiplying 12 by 5. That being said, the perimeter of the pentagon is given by:

= 12 + 12 + 12 + 12 + 12

= 12 × 5

= 60cm

Pentagonal Volume Formula

For the purpose of finding out the volume of a regular pentagonal prism, we would first have to find the apothem length (a). The apothem length is actually a measure from the mid of a polygon to the midpoint of any side. The formula to calculate the volume of a pentagonal prism is as stated:

Volume of pentagonal prism = (5/2) × a × b × h cubic units

In which,

a represents the Apothem length of the pentagonal prism

b represents the Base length of the pentagonal prism

h represents the Height of the pentagonal prism

Solved Example Using Formula For a Pentagon

Example:

If the length of the side of a regular pentagon is equal to 9 cm, then find its perimeter.

Solution:

Given,

length of the side of pentagon = 9 cm

Perimeter of pentagon = 5 x side

= 5 x 9 = 45 cms