Acylinder 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.