Answer:
apothem = 6.9 (nearest tenth)
central angle = 60°
side length = 8
perimeter = 48
area = 96√3 = 166.3 (nearest tenth)
Step-by-step explanation:
Apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides.
⇒ apothem = 4√3 = 6.9 (nearest tenth)
Central angle of a regular polygon = [tex]\dfrac{360}{n}[/tex] (where n is the number of sides)
⇒ central angle = 360 / 6 = 60°
Side length of a regular polygon
apothem = [tex]\dfrac{s}{2tan(\frac{180}{n})}[/tex] (where s is the side length and n is the number of sides)
[tex]\implies 4\sqrt{3}=\dfrac{s}{2tan(\frac{180}{6})}[/tex]
[tex]\implies 4\sqrt{3}=\dfrac{s}{2tan(30)}[/tex]
[tex]\implies s=8\sqrt{3}tan(30)[/tex]
[tex]\implies s=8[/tex]
Perimeter of a regular polygon = s x n
(where s is the side length and n is the number of sides)
⇒ perimeter = 8 x 6 = 48
Area of a regular polygon = [tex]\dfrac12ap[/tex] (where a is the apothem and p is the perimeter)
⇒ area = [tex]\dfrac12 \times 4\sqrt{3} \times 48[/tex]
⇒ area = 96√3 = 166.3 (nearest tenth)
