解:连接$OC.$设$\odot O$的半径为$R.$$\because AB\perp CD,$$\therefore \angle CEO = 90^{\circ}.$在$Rt\triangle OCE$中,由勾股定理,得$OC^{2}=CE^{2}+OE^{2},$即$R^{2}=6^{2}+(R - 2)^{2},$
展开$(R - 2)^{2}$得$R^{2}=36+R^{2}-4R + 4,$
移项可得$4R=40,$
解得$R = 10.$$\therefore \odot O$的半径为$10$