Answer:
Heptagon = 84 ft²
Octagon = 186 yd²
Triangle = 48√3 m²
Step-by-step explanation:
[tex]\textsf{area of a regular polygon}=\dfrac12ap[/tex]
where:
- [tex]a[/tex] = length of apothem
- [tex]p[/tex] = perimeter
Heptagon
a = 5.0 ft
p = 33.6 ft
⇒ area = 1/2 x 5.0 x 33.6 = 84 ft²
Octagon
a = 7.5 yd
p = 8 x 6.2 = 49.6 yd
⇒ area = 1/2 x 7.5 x 49.6 = 186 yd²
Triangle
Formula for the apothem of an equilateral triangle :
[tex]a=\dfrac{s}{2\sqrt{3}}[/tex] where [tex]s[/tex] = side length
a = 4, so:
[tex]4=\dfrac{s}{2\sqrt{3}}[/tex]
[tex]\implies 8\sqrt{3}=s[/tex]
Now we know the side length, we can calculate the perimeter (p):
p = 3 x 8√3 = 24√3
⇒ area = 1/2 x 4 x 24√3 = 48√3 m²
