To create a simple calculator for finding the square root of a number, the method involves using the iterative formula x' = (a/x + x)/2, where 'a' is the input number and 'x' is the current estimate of the square root. This formula is based on the Babylonian method, also known as Heron's method, which improves the estimate with each iteration. The process continues until the estimate converges to a stable value. This approach is efficient and can be easily implemented in a programming environment.