Acone fits inside a square pyramid 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.

since the area of the circle is startfraction pi over 4 endfraction the area of the square, the volume of the cone equals

startfraction pi over 4 endfraction the volume of the pyramid or startfraction pi over 4 endfractionstartfraction pi over 4 endfraction (startfraction (2 r) (h) over 3 endfraction) or one-sixthπrh.
startfraction pi over 4 endfraction the volume of the pyramid or startfraction pi over 4 endfractionstartfraction pi over 4 endfraction (startfraction (2 r) squared (h) over 3 endfraction) or one-thirdπr2h.
startfraction pi over 2 endfraction the volume of the pyramid or startfraction pi over 2 endfraction or two-thirdsπr2h.
startfraction pi over 2 endfraction the volume of the pyramid or startfraction pi over 4 endfraction or one-thirdπr2h.