Question

一球内切与圆锥,已知球和圆锥的的底面半径分别为r,R,求圆锥的体积

Answer 1

设圆心与圆锥底面的边的夹角为α,则圆锥侧面与地面夹角为2α.

tgα=r/R

tg2α=2tgα/[1-(tgα)^2]=(2r/R)/[1-(r/R)^2]=2rR/(R^2-r^2)

圆锥高h=Rtg2α=2rR^2/(R^2-r^2)

圆锥体积V=∏(R^2)h/3=2∏r(R^4)/3(R^2-r^2)