Question 622521
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.
The area of the square is {{{x^2=50}}} square inches.
The height of the cylinder (with base area {{{25pi}}} square inches) is 12 inches, so the volume of the cylinder is
{{{25pi*12=highlight(300pi)}}} cubic inches.
The height of the prism (with base area 50 square inches) is 12 inches, so the volume of the cylinder is
{{{50*12=600}}} cubic  inches.