如图是经过底面最长对角线的正六棱柱的截面,
设正六棱柱上、下底面中心分别为O₁、O₂,
则球心O必定在OO₁的中点,根据题意得底面正六边形的边长为
1/6 × 3 = 1/2,
所以底面积 S
= 6 × √3/4 x (1/2)²
= (3√3)/8
∵该正六棱柱的体积为V
= Sh
= [(3√3)/8] x h
= 9/8,
∴正六棱柱的高 h
= (9/8)/[(3√3)/8]
= √3,
得O至O₁ = √3/2
所以Rt△AOO₁中,AO
=√[(OO₁)²+(AO₁)²]
=1,即外接球的半径R=1
∴正六棱柱的外接球的体积V球
= 4π/3 × R³
= 4π/3
结论,球的体积是 4π/3
蒙采纳~~