Question 593322
<pre>
Here is the regular octagon with one apothem drawn in red:

{{{drawing(400,400,-1.2,1.2,-1.2,1.2,
locate(-.15,sin(11pi/8),12.6), red(line(0,0,0,-sin(13pi/8))),
line(cos(15pi/8),sin(15pi/8),cos(pi/8),sin(pi/8)),
line(cos(3pi/8),sin(3pi/8),cos(pi/8),sin(pi/8)),
line(cos(3pi/8),sin(3pi/8),cos(5pi/8),sin(5pi/8)),
line(cos(5pi/8),sin(5pi/8),cos(7pi/8),sin(7pi/8)),
line(cos(7pi/8),sin(7pi/8),cos(9pi/8),sin(9pi/8)),
line(cos(9pi/8),sin(9pi/8),cos(11pi/8),sin(11pi/8)),
line(cos(11pi/8),sin(11pi/8),cos(13pi/8),sin(13pi/8)),
line(cos(15pi/8),sin(15pi/8),cos(13pi/8),sin(13pi/8))
 )}}}

Area = {{{1/2}}}(apothem)(perimeter) 

The area is given as 770 inē.

The perimeter is 8 times 12.6 in or 100.8 in.

Replacing "Area" by 770, perimeter by 100.8 in

Area = {{{1/2}}}(apothem)(perimeter)

 770 = {{{1/2}}}(apothem)(100.8)

Multiply the {{{1/2}}} by the 100.8 getting 50.4

 770 = 50.4(apothem)

Divide both sides by 50.4:

 {{{770/50.4}}} = apothem

    15.27777778 = apothem

To the nearest whole number 15 inches.

Edwin</pre>