I know there is a method for calculating the square root manually, but that one ins not particularly useful for calculating it on a computer. Computers can multiply and divide numbers with many digits and the execution time is nearly the same, no matter the factor are a single digit or contain many decimal places.
One of the easiest square root algorithms for a computer is the so-called Babylonian method (in fact an application of the Newton algorithm for solving an equation):
x = 0.5 * (x + s/x);
s being the number that you want to take the square root of. See
http://en.wikipedia.org/wiki/Square_root_algorithm[
^].
For the starting value you might choose x=s/4. Four iterations of the above formula should give you a precision of about 6 significant digits.