表面积为6根三的正四面体各个顶点都在同一球面上,则此球的体积为多少

设正四面体OABC的边长为a,球的半径为R.
则
正四面体的表面积 = 4*（1/2）a^2(3^(1/2)/2) = a^2[3^(1/2)] = 6[3^(1/2)],
a = 6^(1/2).
过点O作平面ABC的垂线,垂足为D.
连接AD.
AD = (2/3)(3^(1/2)/2)a = a/3^(1/2) = 2^(1/2)
OD = [a^2 - AD^2]^(1/2) = [6 - 2]^(1/2) = 2.
(OD - R)^2 + AD^2 = R^2,
0D^2 - 2R*OD + AD^2 = 0,
R = [OD^2 + AD^2]/[2*OD] = [4 + 2]/[2*2] = 3/2
球的体积 = (4PI/3)R^3 = (4PI/3)(3/2)^3 = 9PI/2