Answer:
Apothem of the polygon = 24.25 m
Perimeter = 168 m
Area of the polygon = 2,036.89 m²
Step-by-step explanation:
Apothem
The apothem of a regular polygon is the distance from the center of the polygon to the midpoint of any side.
The formula to find the apothem of a regular polygon is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Apothem of a regular polygon}}\\\\a=\dfrac{s}{2 \tan\left(\dfrac{180^{\circ}}{n}\right)}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$s$ is the side length.}\\ \phantom{ww}\bullet\;\textsf{$n$ is the number of sides.}\end{array}}[/tex]
In this case:
Substitute the values into the formula and solve for a:
[tex]a=\dfrac{28}{2 \tan\left(\dfrac{180^{\circ}}{6}\right)}\\\\\\a=\dfrac{28}{2 \tan\left(30^{\circ}\right)}\\\\\\a=\dfrac{14}{\tan\left(30^{\circ}\right)}\\\\\\a=\dfrac{14}{\dfrac{1}{\sqrt{3}}}\\\\\\a=14\sqrt{3}\\\\\\a=24.25\; \sf m\;(nearest\;hundredth)[/tex]
Therefore, the apothem of the given polygon rounded to the nearest hundredth is 24.25 m.
[tex]\dotfill[/tex]
Perimeter
The perimeter of a geometric shape is the sum of the lengths of all its sides.
The perimeter of a regular polygon can be calculated by multiplying the length of one side (s) by the number of sides (n):
[tex]\textsf{Perimeter}=s \cdot n\\\\\textsf{Perimeter}=28 \cdot 6\\\\\textsf{Perimeter}=168\; \sf m[/tex]
Therefore, the perimeter of the given polygon is 168 m.
[tex]\dotfill[/tex]
Area
To find the area of a regular polygon, we can use the following formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Area of a regular polygon}}\\\\A=\dfrac{p\cdot a}{2}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$p$ is the perimeter.}\\ \phantom{ww}\bullet\;\textsf{$a$ is the apothem.}\end{array}}[/tex]
In this case:
Substitute the values into the formula and solve for area:
[tex]A=\dfrac{168 \cdot 14\sqrt{3}}{2}\\\\\\A=84\cdot 14\sqrt{3}\\\\\\A=1176\sqrt{3}\\\\\\A=2036.89\; \sf m\;(nearest\;hundredth)[/tex]
Therefore, the area of the given polygon rounded to the nearest hundredth is 2,036.89 m.
