Answer
1080 $ft^{2}$
Work Step by Step
Given in a regular octagon the approximate ratio of the length of an apothem to the length of a side is 6:5
For a regular octagon with a side of length 15 ft
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 6:5. By proportionality constant x
The apothem a = 6x
side s= 5x
But given side of length = 15 ft
s = 5x = 15ft
x = $\frac{15}{5}$ = 3 ft
apothem of octagon a = 6x = 6* 3ft= 18 ft
The perimeter of octaon P = 8s = 8 * 15 = 120 ft
The area of octagon A = $\frac{1}{2}$ ap
= $\frac{1}{2}$ *18 * 120 = 1080 $ft^{2}$
