Question

What is the area of a hexagon whose perimeter is 24 feet?

Answer 1

Hint: We need to find the area of a hexagon whose perimeter is 24 feet. We start to solve the given question using the formula of the perimeter $P=6\times side$ to find the length of the side of the hexagon. Then, we find the area of the hexagon using the formula $A=\dfrac{3\sqrt{3}}{2}{{\left( side \right)}^{2}}$

Complete step by step solution:
We are given a hexagon of perimeter 24 feet and are asked to find the area of the polygon. We will be solving the given question formula of the perimeter $P=6\times side$ to find the length of the side of the hexagon and then find the area of the hexagon using the formula $A=\dfrac{3\sqrt{3}}{2}{{\left( side \right)}^{2}}$
A polygon is a two-dimensional figure that has a finite number of sides. A hexagon is a polygon with 6 sides.
The hexagon in which the length of all sides and measure of all angles are equal is called a regular hexagon.
The perimeter of the hexagon is the distance around the hexagon. It is given as follows,
$\Rightarrow P=6a$
Here,
P is the perimeter of the hexagon
a is the side length of the hexagon
The area of the polygon is the space or the region that is occupied inside the hexagon. It is given as follows,
$\Rightarrow A=\dfrac{3\sqrt{3}}{2}{{a}^{2}}$
Here,
A is the area of the hexagon
a is the side length of the hexagon
Now, we need to find the side length of the hexagon using the formula of the perimeter of the hexagon.
$\Rightarrow P=6a$
From the question, we know that P = 24 ft
Substituting the value of P in the above equation, we get,
$\Rightarrow 24=6a$
Dividing the above equation with 6 on both sides, we get,
$\Rightarrow \dfrac{24}{6}=\dfrac{6a}{6}$
Simplifying the above equation, we get,
$ a=4ft$
According to the question, we need to find the area of the hexagon.
$\Rightarrow A=\dfrac{3\sqrt{3}}{2}{{a}^{2}}$
From the above, we know that a=4.
Substituting the value of a in the above equation, we get,
$\Rightarrow A=\dfrac{3\sqrt{3}}{2}\times 4\times 4$
Simplifying the above equation, we get,
$\Rightarrow A=\dfrac{3\sqrt{3}}{2}\times 16$
$ A=24\sqrt{3}f{{t}^{2}}$
Substituting the value of $\sqrt{3}$ , we get,
$ A\cong 41.569f{{t}^{2}}$

$\therefore$ The area of a hexagon whose perimeter is 24 feet is approximately equal to 41.569 $f{{t}^{2}}$

Note: Any mistake in writing the formula of the perimeter and area of the hexagon will result in incorrect solution. We must not forget to write the units of the area and side of the hexagon after calculation.