Question

What is the area of a regular pentagon if the apothem is $$4.9$$m and the side is $$7.1$$m?

Answer 1

Hint: A pentagon is a five-sided polygon, also called $5$-gon. It can be regular in shape as well as irregular in shape. In regular, polygon sides and angles are equal to each other, in which the interior angle is equal to $108$ degrees and exterior angle is equal to $72$ degrees. Generally, an angle inside a polygon at the vertex of the polygon is known as an interior angle of a polygon whereas an angle outside a polygon at a vertex of the polygon, formed by one side and the extension of an adjacent side can be called as an exterior angle of a polygon.
Formula used:
The area of the pentagon can be calculated from the below formula,
$$Area=$$$${\displaystyle \frac{1}{2}}\times perimeter\times apothem$$
Where perimeter is equal to the sum of all sides of the polygon,
and apothem is the line from the center of the pentagon to a side such that it intersects the side at $90$ degrees.

               Apothem
Given: Length of the side of the pentagon$$=7.1m$$
             Length of the apothem $$=4.9m$$
To find: Area of the regular pentagon

Complete step by step answer:
Step 1: we know that area of the regular pentagon is given by,
$$Area=$$$${\displaystyle \frac{1}{2}}\times perimeter\times apothem$$
Substituting the given value of perimeter and apothem in the above formula, we get
$$Area=$$$${\displaystyle \frac{1}{2}}\times (5\times 7.1m)\times 4.9m$$ (Perimeter = sum of the side of the pentagon)
Step 2: Now multiplying each term, we get
$$Area=$$$${\displaystyle \frac{1}{2}}\times 35.5m\times 4.9m$$
Further solving, we get
$$Area=$$$$86.975m^2$$
Step 2: Rounding off to two decimal places, we get
$$Area=$$$$86.98m^2$$

Note:
 If we have the only value of the length of the side of the pentagon, then the area of the pentagon can be calculated by using the formula,
 $$Area={\displaystyle \frac{1}{4}}\sqrt{5(5+2\sqrt{5}}{a}^2$$, where $$a$$ is the length of the side of the pentagon.
Also, an apothem is a line from the center of the pentagon to a side such that it intersects the side at $90$ degrees.