Question

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 StartFraction pi r squared Over 4 r squared EndFraction or StartFraction pi Over 4 EndFraction.

A cylinder is inside of a square prism. The height of the cylinder is h and the radius is r. The base length of the pyramid is 2 r.

Since the area of the circle is StartFraction pi Over 4 EndFraction the area of the square, the volume of the cylinder equals

StartFraction pi Over 2 EndFraction the volume of the prism or StartFraction pi Over 2 EndFraction(2r)(h) or πrh.
StartFraction pi Over 2 EndFraction the volume of the prism or StartFraction pi Over 2 EndFraction(4r2)(h) or 2πrh.
StartFraction pi Over 4 EndFraction the volume of the prism or StartFraction pi Over 4 EndFraction(2r)(h) or StartFraction pi Over 4 EndFractionr2h.
StartFraction pi Over 4 EndFraction the volume of the prism or StartFraction pi Over 4 EndFraction(4r2)(h) or Pir2h

Answer 1

Square base side = 2r, so square area = (2r)^2 = 4r^2 and prism volume = 4r^2·h. The cylinder volume = base area of circle · h = πr^2·h. Since πr^2/(4r^2) = π/4, the cylinder is (π/4) of the prism, and its volume = (π/4)(4r^2h) = πr^2h.

So the correct choice is the last one: (π/4) of the prism, i.e. πr^2h.