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Approximating Square Roots

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The ‘square root’ of a given positive number ‘n’ is that number which, when multiplied by itself, gives ‘n’.


If the square root of a given number is not a whole number, then we can approximate the square root of a number.

Square Roots of Perfect Squares

\sqrt{1} 1
\sqrt{4} 2
\sqrt{9} 3
\sqrt{16} 4
\sqrt{25} 5
\sqrt{36} 6
\sqrt{49} 7
\sqrt{64} 8
\sqrt{81} 9
\sqrt{100} 10
\sqrt{121} 11
\sqrt{144} 12


Steps to approximate square roots:

Step 1: Think of two perfect squares between which the given number lies.

Example: If we want to find the square root of 12, then 32 = 9 and 42 = 16. So, \sqrt{12} lies between 9 and 16.

Step 2: Divide the given number by one of the two square roots in step 1.

12 ÷ 3 = 4

Step 3: Find the average of the root and the result in step 2.

\frac{4 + 3}{{2}}=\frac{7}{{2}} =3.5

Step 4: Repeat steps 2 and 3 until you reach the desired result.

Repetition of step 2: 12 ÷ 3.5 = 3.4285

Repetition of step 3: =\frac{3.4285+3.5}{{2}}=\frac{6.9285}{{2}}= 3.46425

Now you need to decide the level accuracy to find the square root of given number. If you want it to be more accurate, keep on repeating steps 2 and 3. We leave \sqrt{12}  here.


Find the square root of the given numbers, rounded to the nearest tenth.

  1.  \sqrt{165}
  2.  \sqrt{189}
  3.   \sqrt{44}
  4.  \sqrt{103}
  5.  \sqrt{114}
Answer key
  1.  12.8
  2.  13.7
  3.  6.6
  4.  10.1
  5.  10.7

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