Estimating a Square Root

The process of estimating a square root to 1 decimal places is quite easy, and. You are simply interpolating between two square numbers.
Suppose you want to estimate the square root of 85.
85 lies between the consecutive square numbers 81 and 100,

\[\frac{85-81}{100-81}=\frac{4}{19} \simeq 0.2105\]

Hence the square root of 85 is approximately 8.2105 or 9.2 to 1 decimal place.
In fact the square root of 85 is 9.21954 to 5 decimal places. Bigger square numbers can be calculates more accurately using this method, and the method is most accurate when estimating a square root that is just a little bigger than a square number.
The error in the above estimate is more than 0.1%. If we were to estimate the square root of 82 using this method the error would only be 0.03%.

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