r=根号下4次方2a^2/3
h=2根号下a^2-r^2显然满足条件的圆柱被经过圆心且平行于底面的平面平分为两部分则圆柱底面积=πr²h=2√(a²-r²)V=πr²*2√(a²-r²)=4π√[(r²/2)²(a²-r²)]根据均值不等式(a²/3)³=[(r²/2+r²/2+a²-r²)/3]³≥(r²/2)²(a²-r²)当r²/2=a²-r²时取等号此时r=√6a/3，h=2√3a/3