How do you estimate a square root in your head?

Start from the perfect squares you already know, ideally through 15 squared. To estimate the square root of 60, note that 7 squared is 49 and 8 squared is 64, so the answer sits between 7 and 8 and much closer to 8 since 60 is near 64. Linear interpolation gives roughly 7.7, and the actual value is 7.746. That level of precision is sufficient for estimating dimensions, distances, and standard deviations in your head, which is what this is for.

A refinement that converges quickly is the Babylonian method, which is straightforward enough to do mentally for one iteration. Guess a value, divide the original number by your guess, then average the two results. For 60 with a guess of 8: 60 divided by 8 is 7.5, and averaging 8 and 7.5 gives 7.75, already accurate to within a hundredth. Repeating the process squares the accuracy each time, which is why it converges so fast and why it has been in use for several thousand years.

Two useful extensions. For large numbers, factor out perfect squares first: the square root of 2000 is the square root of 400 times 5, which is 20 times the square root of 5, and knowing that root 2 is about 1.414, root 3 about 1.732, and root 5 about 2.236 covers most cases. And for numbers just above or below a known square, a quick approximation is to take the known root and add the difference divided by twice that root, so root 51 is approximately 7 plus 2 divided by 14, giving 7.14 against a true value of 7.141. That formula is a first-order Taylor expansion and works well whenever the number is reasonably close to a perfect square.