I have sometimes occupied myself (i.e. in boring meetings or at school prize givings where you can't use paper) by calculating square roots in my head by trial and error i.e. guess the root and square it, then if the answer is too high, try again with a slightly smaller guess etc.
D'oh, I am so dumb.
There is a much simpler, quicker and more obvious method. Which most people probably already know, but here it is for the record.
If you have to guess √28, you start with the nearest known square number i.e. 25, imagine a square (illustrated below) with side length 5 = area 25 and overlay it onto a square with area 28. The total surface area of the gray shaded cells = 3 (28 minus 25). There are ten such cells (plus a smaller square in the bottom right, which I'll get to later), so each cell has area of a smidge less 0.3. They are one unit high (or wide) so the width (or height) is a smidge less than 0.3.
So as a first approximation, √28 = 5.3.
This overstates the answer slightly, because of the smaller square (5.3^2 = 28.09). We can do the same process again - divide the area of the smaller square (0.3^2 = 0.09) by ten = 0.009 and deduct that from 5.3 = 5.291.
Checks on calculator: 5.291^2 = 27.995. Close enough.
The same applies if the nearest square number is larger. So first approximation = √80 = 9 - (1/18) = 8.944. When you are this close, there is no point bothering with reapportioning the smaller square (again, you subtract it from the first approximation), because its area is only 0.056^2, and 0.056^2 ÷ 18 = so close to zero as makes no difference (and mighty difficult to calculate in your head, it's 0.00017).
Checks on calculator: 8.944^2 = 80.003. Again, close enough.
Here's the diagram, in case my explanation is not clear: