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,
\[Area = \]\[\dfrac{1}{2} \times perimeter \times apothem\]
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,
\[Area = \]\[\dfrac{1}{2} \times perimeter \times apothem\]
Substituting the given value of perimeter and apothem in the above formula, we get
\[Area = \]\[\dfrac{1}{2} \times (5 \times 7.1m) \times 4.9m\] (Perimeter = sum of the side of the pentagon)
Step 2: Now multiplying each term, we get
\[Area = \]\[\dfrac{1}{2} \times 35.5m \times 4.9m\]
Further solving, we get
\[Area = \]\[86.975{m^2}\]
Step 2: Rounding off to two decimal places, we get
\[Area = \]\[86.98{m^2}\]

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,
 \[Area = \dfrac{1}{4}\sqrt {5(5 + 2\sqrt 5 } {a^2}\], 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.