Question 732663
{{{drawing(300,400,-4.5,4.5,-5,7,
circle(0,0,4),
line(4,0,-4,0),line(0,7,0,-4),
red(line(-4,0,0,-4)),red(line(0,-4,4,0)),
circle(4,0,8),circle(-4,0,8),
green(circle(0,4,2.343)),
line(-4,0,4,8),line(4,0,-4,8),
red(line(-4,0,-1.657,5.657)),red(line(4,0,1.657,5.657)),
red(line(-1.657,5.657,0,6.343)),red(line(0,6.343,1.657,5.657)),
locate(-0.1,-4.1,D),locate(-4.3,0.2,A),locate(4.1,0.2,B),
locate(0.1,-0.1,O),locate(0.1,3.7,C),
locate(1.8,5.9,P),locate(-2,5.9,Q),locate(0.1,6.2,R)
)}}} I did not know how to draw just the arc PQR, so I had to draw the whole green circle.
The sides of the square ADBC are {{{4sqrt(2)=about}}}{{{5.657feet}}} in length.
{{{CD=AB=AP=BQ=8}}} because they are radii of the circles containing the arcs BP and AQ
so {{{CR=CP=CQ=8-4sqrt(2)}}} all radii of my green circle containing arc PRQ
 
So {{{DR=CD+CR=8+8-4sqrt(2)=16-4sqrt(2)=about}}}{{{10.343feet}}}
 
BPA and QAB are isosceles triangles with a {{{45^o}}} vertex angle and legs measuring 8 feet.
Based on law of cosines or  using the fact that BPC and AQC are right triangles, we can calculate that {{{BP^2=AQ^2=64(2-sqrt(2))=about}}}{{{37.49feet^2}}}
The approximate length would be {{{BP=AQ=sqrt(64(2-sqrt(2)))=8sqrt(2-sqrt(2))=about}}}{{{6.123feet}}}
Otherwise we could split those triangles into two congruent right triangles with a {{{22.5^o}}} angle and 8-foot hypotenuse, and calculate the length of their short legs (in feet) as {{{BP/2=AQ/2=8sin(22.5^o)=about}}}{{{8*0.3827=3.0615}}}
Either way the ratio of base to leg length in those isosceles triangles is {{{sqrt(2-sqrt(2))=about}}}{{{0.7654}}}
 
PRC and RQC are also isosceles triangles with a {{{45^o}}} vertex angle, so they are similar to BPC and AQC.
We knew that the length of their legs (in feet) were
{{{CR=CP=CQ=8-4sqrt(2)}}} and multiplying that times the ratio found above for the similar triangles we can find the length of {{{PR=RQ}}}.
Giving up on accurate value expressions,
{{{CR=CP=CQ=8-4sqrt(2)=about}}}{{{2.343}}},
so {{{PR+PQ=about}}}{{{1.793feet}}}
 
Now we can calculate the perimeter of ADBPRQ as the approximate value (in feet) of
{{{5.657+5.657+6.123+6.123+1.793+1.793=27.146}}}