Learn How To Find The Square Root Of A Number Using the Estimation Method

What’s a Square Root Anyway? Review

Math classes often include a unit on how to find the square root of a number. The concept of the square root is related to other mathematical ideas and concepts such as functions, graphing and trigonometry. Finding the square root of a number can also be used in practical situations like construction projects and work projects that involve calculating angles. In mathematics, the square root can be seen as the reverse of squaring a number (e.g., the square of 2 is 4 and the square of 10 is 100).

Square Root (Image Credit: Wikimedia Commons)

This article will explain the estimation method of finding the square root of a number by hand without the use of a calculator or computer. Some methods are more precise and time consuming than others. If you have to find square roots in a math homework assignment, check with your teacher to see if you are required to use a particular method.

Note: The image to the left shows the square root symbol: √. This symbol should be used whenever you are solving square root problems.

Finding Perfect Square Roots

Some numbers are called perfect squares since they can be found easily. Here are some examples to illustrate the concept:

The √100 is 10 (i.e. 10 multiplied by 10 equals 10).

The √81 is 9

The √64 is 8.

The √49 is 7.

The √36 is 6.

The √25 is 5.

Using the above examples, one can easily make a table of perfect squares. Many numbers do not have such a clear answer, however, and for those numbers, different methods are needed.

The Estimation Method

The estimation method of finding square roots is a good way to find an approximate answer. This approach requires a bit of trial and error as well as a basic knowledge of perfect square root numbers. This section provides several examples to show you how to find the square root of a number using the estimation method.

Example 1: What is √50?

Step 1: What perfect square roots are close to 50?

Step 2: The nearest perfect square roots to 50 are 7 and 8.

Step 3: The √ of 49 is 7, we can estimate that the √ of 50 is likely closer to 7 than 8.

Step 4: The estimated √50 is slightly more than 7 (in fact, the √ of 50 is about 7.07!)

Example 2: What is √19?

Step 1: What perfect square roots are close to 19?

Step 2: The nearest perfect square root of 16 (i.e. 4 times 4 is 16); the second closest perfect square is 25 (i.e. 5 times 5 is 25).

Step 3: The √ of 16 is (i.e. 4) much closer to 19 than the √ of 25 (i.e. 5)

Step 4: The estimated √19 is between 4 and 4.5 (the actual square root of 19 is 4.35)

Other Methods

The estimation method works well with relatively small numbers. However, if you are trying to find the square of a larger number such as 500 or 5,000, you may find the estimation method difficult to use. In that case, you can use a variant of the estimation method that lets you calculate to the second decimal place or further. Alternately, you can consider using a calculator.


  • Image: Wikimedia Commons/David Vignoni, http://commons.wikimedia.org/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg
  • Math: Square Root, http://ethemes.missouri.edu/themes/12
  • Calculating Square Roots, http://www.math.toronto.edu/mathnet/questionCorner/sqrootcalc.html