Question 1181238


we need a diagonals across 6 sides ({{{d[6]}}}), {{{5}}} sides ({{{d[5]}}}), {{{4}}} sides ({{{d[4]}}}), {{{3}}} sides({{{d[3]}}}) , and {{{2}}} sides ({{{d[2]}}})

{{{d[6]=(sqrt(6)+sqrt(2))*a}}} ........given {{{a=2cm}}}
{{{d[6]=(3.863703305156273)*2cm}}}
{{{d[6]=7.727cm}}}
{{{d[5] = ( 2 + sqrt(3) ) * 2=7.464cm}}}
{{{d[4] = (( 3*sqrt(2) + sqrt(6) ) / 2) * 2=6.692cm}}}
{{{d[3] = ( sqrt(3) + 1 ) * 2=5.464cm}}}
{{{d[2 ]= (( sqrt(6)+sqrt(2) ) / 2)* 2= d[6] / 2=3.864cm}}}

then radius of circumcircle is:

{{{r[c] = d[6] / 2 = d[2]=3.864cm}}}
{{{r[c] = 3.864cm}}}

{{{A[cc]=(3.864cm)^2*pi}}} ...........where {{{cc}}} stands for circumcircle 
{{{A[cc]=14.93*pi*cm^2}}}

radius of incircle  is:
{{{r[i]= d[5] / 2=7.464/2=3.732cm}}}

{{{r[i]=3.732cm}}}

{{{A[ic]=(3.732cm)^2*pi}}}...........where {{{ic}}} stands for incircle 
{{{A[ic]=13.93*pi*cm^2}}}

 the difference in area between the inscribed circle and the circumscribed circle in a 12-gon is

{{{A[cc]-A[ic]=14.93*pi*cm^2-13.93*pi*cm^2=pi}}}