Question 909337
R, radius of the outer circle
r, radius of the inner (smaller) circle
{{{R=8/2=4}}} inches
r is unknown.


Area of outer circle, {{{pi*R^2}}};
Area of smaller inner circle, {{{pi*r^2}}};
Difference in areas is the area of your shaded ring surrounding the inner circle, {{{pi*R^2-pi*r^2}}}.


According to the description, {{{pi(R^2-r^2)=pi*r^2}}}.
You want to solve for r.


STEPS
Divide both members by pi.
{{{R^2-r^2=r^2}}}
{{{R^2=2r^2}}}
{{{r^2=(1/2)R^2}}}
{{{r=R/sqrt(2)}}}, and you might want to rationalize the denominator.
{{{r=(R/sqrt(2))(sqrt(2)/sqrt(2))}}}
{{{highlight(r=R*sqrt(2)/2)}}}----the symbolic answer.


Substitute for R and evaluate r.