Abstract: In this paper, we introduce the concepts of cut ratio and covered ratio of domains. Further, we give an explicit estimation of the constant in the general Ahlfors' covering theorem and Ahlfors' fundamental theorem on a sphere. Because of the importance of these theorems, it is no doubt that these rstimations is significant. Applying these yesults, we improve a series of theorems, prove that there exists a sequence of filling circles for algebroidal function. and establish a new normal criterion. (These works will be done in other papers.)