I was recently studying AC circuits and there I need to use Pythagoras theorem a lot.So I was looking for a method with which square roots can be calculated very fast,manually up to 1 decimal place of accuracy.Is there any such method?
For example say you have to find the square root of $322\pi$ upto $1$ decimal place of accuracy in about 2 mins without a calculator.What method would you follow?
I use the square root algorithm for a general case, quite similar to the long division. You can look at how and why the algorithm works here.
Newton's method is also a very fast way to quickly approximate the square root of a number.
For special cases, you can try to keep square roots of certain primes memorized and then multiply the square roots of the prime factors of the number in consideration. This works for most cases for me.
Example: $\sqrt {10} = \sqrt 2 \times \sqrt 5$. $2$ and $5$ are primes, the square root of whom you have memorized. Now you reduced the square root problem to a multiplication problem, which is much faster to compute.