三棱锥pabc中pa垂直平面abc,ab垂直bc,pa=ab=bc＝2,则内切球的体积为

假设内切球体的圆心为O,内切球体的半径为R,
∵pa垂直平面abc,ab垂直bc,
∴BC⊥PB PA⊥AC
又∵pa=ab=bc＝2
∴PB=√（PA²+AB²）=√（4+4）=2√2
AC=√（AB²+BC²）=√（4+4）=2√2
根据S P-ABC=1/3×PA×S△ABC=S O-PAB+S O-PAC+S O-PBC+S O-ABC
=1/3×R×(S△PAB+S△PAC+S△PBC+S△ABC)
=1/3×R×(1/2×PA×AB+1/2×PA×AC+1/2×PB×PC+1/2×AB×AC)
=1/3×R×(1/2×2×2+1/2×2×2√2+1/2×2√2×2+1/2×2×2）
=1/3×R×(4+4√2）
=1/3×2×1/2×2×2
解得R=4/（4+4√2）=√2-1
所以内切球体的体积S=4/3×π×R³=4/3×π×（√2-1）³=4（5√2-7）π/3