Answer:
[tex]\sf Apothem(a) = 24.25 \, \textsf{m}[/tex]
[tex]\sf Perimeter (P)168 \, \textsf{m}[/tex]
[tex]\sf Area(A) = 2036.89 \, \textsf{m}^2[/tex]
Step-by-step explanation:
To calculate the apothem, perimeter, and area of a regular polygon with side length [tex]\sf 28 \, \textsf{m}[/tex] and [tex]\sf 6[/tex] sides, we can use the following formulas:
Apothem ([tex]\sf a[/tex]) of a regular polygon:
[tex]\large\boxed{\boxed{\sf a = \dfrac{s}{2 \times \tan\left(\dfrac{\pi}{n}\right)}}} [/tex]
Where:
- [tex]\sf s[/tex] is the length of one side
- [tex]\sf n[/tex] is the number of sides
Perimeter ([tex]\sf P[/tex]) of a regular polygon:
[tex]\large\boxed{\boxed{\sf P = n \times s}} [/tex]
Where:
- [tex]\sf n[/tex] is the number of sides
- [tex]\sf s[/tex] is the length of one side
Area ([tex]\sf A[/tex]) of a regular polygon:
[tex]\large\boxed{\boxed{\sf A = \dfrac{1}{2} \times a \times P }}[/tex]
Where:
- [tex]\sf a[/tex] is the apothem
- [tex]\sf P[/tex] is the perimeter
Given:
- Side length ([tex]\sf s[/tex]) = [tex]\sf 28 \, \textsf{m}[/tex]
- Number of sides ([tex]\sf n[/tex]) = [tex]\sf 6[/tex]
Let's calculate:
Apothem ([tex]\sf a[/tex]):
[tex]\sf a = \dfrac{28}{2 \times \tan(\dfrac{\pi}{6})} [/tex]
[tex]\sf a = \dfrac{28}{2 \times \tan(30^\circ)} [/tex]
[tex]\sf a = \dfrac{28}{2 \times 0.5773502691896} [/tex]
[tex]\sf a = \dfrac{28 }{2 \times \sqrt{3}} [/tex]
[tex]\sf a = \dfrac{28}{1.1547005383792} [/tex]
[tex]\sf a \approx 24.248711305964 [/tex]
[tex]\sf a \approx 24.25 \, \textsf{m (in nearest hundredth)} [/tex]
Perimeter ([tex]\sf P[/tex]):
[tex]\sf P = 6 \times 28 [/tex]
[tex]\sf P = 168 \, \textsf{m} [/tex]
Area ([tex]\sf A[/tex]):
[tex]\sf A = \dfrac{1}{2} \times 24.248711305964 \times 168 [/tex]
[tex]\sf A = \dfrac{1}{2} \times 48.497 \times 168 [/tex]
[tex]\sf A = 2036.8917497009 [/tex]
[tex]\sf A \approx 2036.89 \, \textsf{m$^2$ (in nearest hundredth)}[/tex]
In summary:
[tex]\sf Apothem(a) = 24.25 \, \textsf{m}[/tex]
[tex]\sf Perimeter (P)168 \, \textsf{m}[/tex]
[tex]\sf Area(A) = 2036.89 \, \textsf{m}^2[/tex]
