Question 121083
You don't specify whether you want the lateral surface area or the total surface area:


For the <b><i>lateral</b></i> surface area:


{{{A=2(pi)rh}}}.


For {{{h = 1}}} and {{{A=12(pi)}}},  this becomes


{{{2(pi)r=12(pi)}}}


Divide both sides by {{{2(pi)}}}


{{{r=6}}}


Therefore, if the lateral surface area is {{{12(pi)}}}, then the radius is 6


For the <b><i>total</b></i> surface area:


{{{A=2(pi)r^2+2(pi)rh}}}, again substituting {{{h = 1}}} and {{{A=12(pi)}}}, we get:


{{{2(pi)r^2+2(pi)r=12(pi)}}}


Again, divide by {{{2(pi)}}}


{{{r^2+r=6}}}


{{{r^2+r-6=0}}}


Factor


{{{(r+3)(r-2)=0}}}, so {{{r=-3}}} or {{{r=2}}}.


We can exclude {{{r=-3}}} because we are looking for a positive number measurement of length.


Therefore if the total surface area is {{{12(pi)}}}, the radius is {{{2}}}