A cylinder fits inside a square prism as shown. For every cross section, the ratio of the area of the circle to the area of the square is or .Since the area of the circle is the area of the square, the volume of the cylinder equals the volume of the prism or (2r)(h) or πrh. the volume of the prism or (4r2)(h) or 2πrh. the volume of the prism or (2r)(h) or r2h. the volume of the prism or (4r2)(h) or r2h.

"For every cross section, the ratio of the area of the circle to the area of the square is or ."

To find this ratio we need to find areas of the circle and the square.
Circle:
[tex]Area= radius^{2} * \pi \\ A_{1} = r^{2} * \pi [/tex]
Square:
[tex]Area= side^{2} \\ A_{2} = (2r)^{2} \\ A_{2} 4r^{2}= [/tex]
Now we divide these two areas to find ratio:
[tex]ratio= \frac{ A_{1} }{ A_{2} } \\ ratio= \frac{r^{2} * \pi }{4r^{2}} \\ ratio= \frac{ \pi }{4} [/tex]

"Since the area of the circle is the area of the square,"
From the ratio above we can see that areas are not same.

"the volume of the cylinder equals"
Formula for volumes of cylinder and prism follow the formula:
[tex]Volume=base*height[/tex]
For cylinder:
[tex]V=r^{2} * \pi *h[/tex]
For prism:
[tex]V= (2r)^{2} \\ V=4 r^{2} [/tex]