Question 540899
{{{drawing(300,300, -1.25,1.25,-1.25,1.25,

line(cos(pi/10),sin(pi/10),0,1), 
line(cos(9pi/10),sin(9pi/10),0,1),
circle(0,0,1), locate(0,-.65,8),
line(cos(9pi/10),sin(9pi/10),cos(13pi/10),sin(13pi/10)),
line(cos(17pi/10),sin(17pi/10),cos(13pi/10),sin(13pi/10)),
line(cos(pi/10),sin(pi/10),cos(17pi/10),sin(17pi/10)) )}}}
<pre>
We draw in 2 lower adjacent radii, of length r. The central angle between 
two adjacent radii subtending a side of the pentagon are {{{"360°"/5}}} or 72°
each:
{{{drawing(300,300, -1.25,1.25,-1.25,1.25,
locate(-.1,-.14,"72°"), locate(-.38,-.32,r),
line(cos(pi/10),sin(pi/10),0,1), 
line(cos(9pi/10),sin(9pi/10),0,1),
circle(0,0,1), locate(0,-.65,8), green(line(0,0,cos(13pi/10),sin(13pi/10)),
line(0,0,cos(17pi/10),sin(17pi/10))),
line(cos(9pi/10),sin(9pi/10),cos(13pi/10),sin(13pi/10)),
line(cos(17pi/10),sin(17pi/10),cos(13pi/10),sin(13pi/10)),
line(cos(pi/10),sin(pi/10),cos(17pi/10),sin(17pi/10)) )}}}

Next we draw in a perpendicular to the bottom side,
(called an apothem) which divides the 8 inch bottom side into two
4 inch segments, and also divides the 72° central angle into two
36° angles:  

{{{drawing(300,300, -1.25,1.25,-1.25,1.25,
locate(-.2,-.25,"36°"), locate(-.38,-.32,r),
line(cos(pi/10),sin(pi/10),0,1), 
line(cos(9pi/10),sin(9pi/10),0,1),
circle(0,0,1), locate(-.3,-.65,4), green(line(0,0,cos(13pi/10),sin(13pi/10)),
line(0,0,0,-sin(3pi/10)),
line(0,0,cos(17pi/10),sin(17pi/10))),
line(cos(9pi/10),sin(9pi/10),cos(13pi/10),sin(13pi/10)),
line(cos(17pi/10),sin(17pi/10),cos(13pi/10),sin(13pi/10)),
line(cos(pi/10),sin(pi/10),cos(17pi/10),sin(17pi/10)) )}}}

Now looking at the right triangle on the left, we have

sin(36°) = {{{opposite/hypotenuse}}} = {{{4/r}}}

Multiply both sides by r

r·sin(36°) = 4

Divide both sides by sin(36°)

r = {{{4/sin("36°")}}}

r = {{{4/.5877852523}}} 

r = 6.905206467 inches

and the diameter is twice the radius, so

diameter of the circle = 2(6.905206467) = 13.61041293 inches

Edwin</pre>