Question

设A,B,C,D为球O上四点,若AB,AC,AD两两互相垂直,且AB=AC=√6,若AD=R(R为球O的半径)则三棱锥D-ABC的体积为

答案为2

Answer 1

AD垂直平面ABC,为三棱锥D-ABC的高,

 

AD=R,设E为BC中点,AE=BE=EC=1/2BC=1/2√AB²+AC²=1/2*2√3=√3,
AE通过球心O,延长AE交球面F,AF=2R,,设EF=X,则有,BE²+(R-x)²=R²,   (√3)²+(R-x)²=R²,

DA垂直AE,则有AE²+DA²=DE²,(√3)²+R²=(2R-x)²,3R²=36,R=2√3

S三角形ABC=1/2AB*AC=1/2*6=3,

V三棱锥D-ABC的体积=1/3*S*AD=1/3*3*2√3=2√3,

追问

由图像AE不过圆心

Answer 2