Question 622471
I assume that from above you just see the bases of the prism and the cylinder, looking like a square inscribed in a circle, like this:
{{{drawing(200,200,-6,6,-6,6,
circle(0,0,5),
rectangle(-3.535,-3.535,3.535,3.535),
locate(-0.5,0.5,10),
arrow(0.5,0,5,0),arrow(-0.5,0,-5,0),
locate(-0.5,-2,5sqrt(2)),
arrow(0.8,-2.5,3.535,-2.5),arrow(-0.8,-2.5,-3.535,-2.5)
)}}} The diameter of the circle is 10 inches, the length of the diagonal of the square.
Two sides of the square, of length x inches, form a right triangle with the diagonal for a hypotenuse, so
{{{x^2+x^2=100}}} --> {{{2x^2=100}}} --> {{{x^2=50}}} --> {{{x=sqrt(50)=sqrt(25*2)=sqrt(25)*sqrt(2)=5sqrt(2)}}}
The area of the circle (with radius 5) is {{{pi*5^2=25pi}}} square inches.
That circle is one of the two congruent (equal) bases of the cylinder.
The circumference is {{{2*pi*5=10pi}}} inches, and the lateral area of the cylinder is that times the height, or
{{{10pi*12=120pi}}} square inches.
So the total surface area of the cylinder is
{{{120pi+2*25pi=170pi}}} square inches.
The area of the square is {{{x^2=50}}} square inches.
That square is one of the two congruent (equal) bases of the prism.
The perimeter of the square is {{{4*5sqrt(2)=20sqrt(2)}}} inches, and the lateral area of the prism is that times the height, or
{{{20sqrt(2)*12=120sqrt(2)}}} square inches.
So the total surface area of the cylinder is
{{{120sqrt(2)+2*50=120sqrt(2)+100}}} square inches.