Quick and Accurate Estimation of Square Roots

Plus: Table of Squares up to 200²

There are several ways to calculate square roots by hand to any degree of accuracy that you desire. Common methods are the Babylonian algorithm and the long division technique.

Unfortunately, these methods are time-consuming to carry out manually. If you don't have a calculator with a square root function and you need to estimate a radical, you can apply the technique below. This trick is always accurate to at least the tenths place, and it increases in accuracy when you apply it to larger numbers.

Step 1: First, call the number whose square root you want to estimate N. Then, identify the perfect squares on either side of N and call the square root of the lesser one B.

For example, suppose you want to estimate the square root of 500. Since 484 = 22² and 529 = 23², you have N = 500 and B = 22. This tells you that the square root of 500 is somewhere between 22 and 23.

Step 2: To figure out where the square root lies in between the two bounds, compute the ratio

[N - B²]/[(B+1)² - B²] = [N - B²]/[2B + 1]

and round to the nearest tenth or hundredth. Using N = 500 and B = 22 in this example, you obtain

[500 - 22²]/[2*22 + 1] = 16/45 ≈ 0.36 or 0.4.

Step 3: Add this number to B to obtain the estimate of your square root. Continuing with the example above, you would estimate the square root of 500 as either 22.36 or 22.4.

As it turns out, the square root of 500 is about 22.360679775, so this estimation technique is accurate enough for most applications.

If you need to review perfect squares, use the table below.

© Had2Know 2010

How to Compute Square Roots by Hand

How to Compute Cube Roots by Hand

Long Multiplication Calculator

Babylonian (Divide-and-Average) Algorithm for Square Roots

Babylonian Algorithm for Computing Cube Roots Without a Calculator

Quick Estimation of Cube Roots

Printable Tables of Logs, Trig, Inverse Trig, and Radicals