Answer
460.8$ft^{2}$
Work Step by Step
Given in a regular dodecagon the approximate ratio of the length of an apothem to the length of a side is 15:8
The length of the apothem = 12ft
The area of a regular polygon whose apothem has length a and whose perimeter P is given by A = $\frac{1}{2}$ ap
Lets take the ratio of apothem and side ratio 15:8. By proportionality constant x
The apothem a = 15x
side s= 8x
But given apothem of length= 12 ft
a = 15x = 12ft
x = $\frac{12}{15}$ = 0.8 ft
side of dodecagon s = 8x = 8 * 0.8 ft= 6.4 ft
The perimeter of dodecagon P = 12s = 12 * 6.4 ft = 76.8 ft
The area of octagon A = $\frac{1}{2}$ ap
= $\frac{1}{2}$ *12 * 76.8 = 460.8$ft^{2}$
