We study the square root computation by Leonardo Fibonacci (or Leonardo of Pisa) in his MSS Liber Abaci from c1202 and c1228 and De Practica Geometrie from c1220. In this MSS, Fibonacci systematically describes finding the integer part of the square root of an integer in numerous examples with three to seven decimal digits. The results of these examples are summarized in a table in the paper. Liber Abaci also describes in detail finding an approximation to the fractional part of the square root. However, in other examples in Liber Abaci and De Practica Geometrie, only the approximate values of the fractional part of the square roots are stated. This paper further explores these approximate values using techniques like reverse engineering. Contrary to many claims that Fibonacci also used other methods or approximations, we show that all examples can be explained using one digit-by-digit method to compute the integer part of the square root and one approximation scheme for the fractional part. Further, it is shown that the approximation scheme is tied to the method to compute the integer part of the square root.