Answer:
[tex]\textsf{A)}\quad \textsf{Side length}=\dfrac{16\sqrt{3}}{3}\; \sf m \approx 9.24\; m[/tex]
[tex]\textsf{B)}\quad \textsf{Perimeter}=32\sqrt{3}\; \sf m \approx 55.43 \; m[/tex]
[tex]\textsf{C)}\quad \textsf{Area}=128\sqrt{3}\; \sf m \approx 221.70\; m[/tex]
Step-by-step explanation:
The formula for calculating the interior angle of a regular polygon with n sides is:
[tex]\textsf{Interior angle}=\dfrac{(n - 2)180^{\circ}}{n}[/tex]
The given regular polygon has 6 sides, so n = 6:
[tex]\textsf{Interior angle}=\dfrac{(6 - 2)180^{\circ}}{6}=\dfrac{720^{\circ}}{6}=120^{\circ}[/tex]
This means that the interior angle of the hexagon is 120°.
Note: An interior angle of 108° indicates that the polygon is a regular pentagon with 5 sides. Assuming that the given diagram of the hexagon is accurate, I have answered the question using the correct interior angle of 120°.
[tex]\hrulefill[/tex]
Part A
The length of the side of the hexagon can be found by using trigonometry.
If we draw a line segment from the center of the hexgon to a vertex, we create a right triangle, where the base is equal to half the side length, the height is equal to the apothem, and the angle between the base and the hypotenuse is half an interior angle.
To find the length of the base of the triangle, we can use the tangent ratio.
[tex]\boxed{\begin{array}{l}\underline{\textsf{Tangent trigonometric ratio}}\\\\\sf \tan(\theta)=\dfrac{O}{A}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{$O$ is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{$A$ is the side adjacent the angle.}\end{array}}[/tex]
In this case:
- The angle is half the interior angle, so θ = 60°.
- The side opposite the angle is the apothem, so O = 8.
- The side adjacent the angle is half the side length, so A = s/2.
Substitute these values into the tangent ratio:
[tex]\tan 60^{\circ}=\dfrac{8}{\frac{s}{2}}[/tex]
Now, solve for s:
[tex]\dfrac{s}{2}=\dfrac{8}{\tan 60^{\circ}}\\\\\\\\s=\dfrac{8\cdot 2}{\sqrt{3}}\\\\\\\\s=\dfrac{16}{\sqrt{3}}\\\\\\\\s=\dfrac{16\cdot \sqrt{3}}{\sqrt{3}\cdot \sqrt{3}}\\\\\\\\s=\dfrac{16\sqrt{3}}{3}\; \sf m[/tex]
Therefore, the length of the side of the hexagon is exactly (16√3)/3 m, which is approximately 9.24 m (rounded to the nearest hundredth).
[tex]\hrulefill[/tex]
Part B
The perimeter of a polygon is the sum of the lengths of all its sides.
Therefore, as the side lengths of a regular polygon are congruent, we can simply multiply the length of one side by 6:
[tex]\textsf{Perimeter}=6s\\\\\\\textsf{Perimeter}=6\cdot \dfrac{16\sqrt{3}}{3}\\\\\\\textsf{Perimeter}=\dfrac{96\sqrt{3}}{3}\\\\\\\textsf{Perimeter}=32\sqrt{3}\; \sf m\\\\\\[/tex]
So, the perimeter of the hexagon is 32√3 meters, which is approximately 55.43 m (rounded to the nearest hundredth).
[tex]\hrulefill[/tex]
Part B
The formula for the area of a regular polygon is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Area of a regular polygon}}\\\\A=\dfrac{ap}{2}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$A$ is the area.}\\\phantom{ww}\bullet\;\textsf{$a$ is the apothem.}\\\phantom{ww}\bullet\;\textsf{$p$ is the perimeter.}\end{array}}[/tex]
In this case:
Therefore:
[tex]\textsf{Area}=\dfrac{8 \cdot 32\sqrt{3}}{2}\\\\\\\textsf{Area}=\dfrac{256\sqrt{3}}{2}\\\\\\\textsf{Area}=128\sqrt{3}\; \sf m[/tex]
So, the area of the hexagon is 128√3 meters, which is approximately 221.70 m² (rounded to the nearest hundredth).
