已知正三棱锥P-ABC,作侧面斜高PM,高PH,
S△PAB=2√3a^2/3=PM*AB/2=PM*a/2,
∴PM=4√3a/3,
∵H是正△ABC的外心,M是AB中点,
∴C\H\M三点共线,
CM=√3AB/2=√3a/2,
HM=CM/3=√3a/6,
根据勾股定理,
PH=√(PM^2-HM^2)=√(16a^2/3-a^2/12)=√21a/2,
VP-ABC=S△ABC*PH/3=(√3a^2/4)*(√21a/2)/3
=√7a^3/8.
∴正三棱锥体积为√7a^3/8.