Answer:
- a = 8.26 ft
- P = 60 ft
- Area = 247.7 ft²
Explanation:
You want the apothem, perimeter, and area of a regular pentagon with side length 12 ft.
Apothem
The apothem is one leg of the right triangle that is half of one of the sectors. The other leg is half the side length. This gives ...
tan(36°) = (6 ft)/a
a = (6 ft)/tan(36°) ≈ 8.25829 ft
Perimeter
The perimeter is simply 5 times the side length:
(12 ft) × 5 = 60 ft
Area
The area is given by the formula ...
A = 1/2Pa
A = 1/2(60 ft)(8.25829 ft) ≈ 247.749 ft²
Summary:
- a = 8.26 ft
- P = 60 ft
- Area = 247.7 ft²
__
Additional comment
An n-sided regular polygon with side length s has an area of ...
A = [s²n]/[4tan(180°/n)]
For s=12 and n=5, this is ...
A = 12²·5/(4·tan(180°/5)) = 180/tan(36°) ≈ 247.7 . . . . square feet