Estimating square roots in your head
To find √n, start from the nearest perfect square a², and add the difference divided by 2a:
√n ≈ a + (n − a²) / (2a)
For √40: the nearest square is 36, so 6 + 4/12 ≈ 6.33. The true value is 6.3246.
For √90: use 81, so 9 + 9/18 = 9.5. The true value is 9.4868.
Where it comes from
This is one step of Newton's method, or equivalently the tangent line of √x at the perfect square.
The estimate always comes out slightly high, because the curve bends below its tangent.
If you need more precision, repeat the step with your estimate as the new a:
a = 6.333… → 6.333 + (40 − 40.111) / 12.667 ≈ 6.3246
Two steps are already better than most people need.