Finding square roots of numbers that aren't perfect squares without a calculator

There are a number of ways to calculate square roots without a calculator. Here is guess and divide method.

Step–1: Estimate/find a perfect square root as close as possible to your number.

Step–2: Divide your number by the square root.

Step–3: Calculate the average of the result of step 2 and the root.

Step–4: Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.

Calculate the square root of 10 (√ 10) to 4 decimal places.

1.    Find the perfect square number closer to 10. 3² = 9 and 4² = 16, so take 3.

2.    Divide 10 by 3. 10÷3 = 3.33 (you can round off the answer)

3.    Average 3.33 and 3. (3.33 + 3)÷2 = 3.1667

Repeat step 2: 10÷3.1667 = 3.1579

Repeat step 3: Average 3.1579 and 3.1667.

(3.1579 + 3.1667)÷2 = 3.1623

Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001

If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3.

Here's another example, √695 =?

1.  Let us first guess for the root as 25.
2. 695 ÷ 25 = 27.8
3. (25 + 27.8) ÷ 2 = 26.4
4. 695 ÷ 26.4 = 26.3257
5. (26.4 + 26.3257) ÷ 2 = 26.36285
6. 695 ÷ 26.36285 = 26.3628553

Last two findings are about equal, so you now have a good estimate of the root of 695. You can keep going through the process until you are satisfied with the accuracy.

So, √24.6=?

1. Let's try 5 since 5² = 25, which is pretty close to 24.6.
2. Divide 24.6 by 5. 24.6 ÷ 5 = 4.92
3. (5 + 4.92) ÷ 2 = 4.96

You can stop here. 4.96 is pretty close to 4.9598 which is the actual square root of 24.6. Repeat steps 2 and 3 to any desired level of accuracy. The further you go, the harder the long division becomes. But the first few cycles yield a pretty close answer.

• 24.6 / 4.96 = 4.9596
• (4.96 + 4.9596) ÷ 2 = 4.9598

Some more examples solved concisely,

√2613 =?

√2500 = 50

2613 ÷ 50 = 52.26

(52.26 + 50) ÷ 2 = 51.13 (approx. answer)

√6673 =?

√6400 = 80

6673 ÷ 80 = 83.41

(83.41 + 80) ÷ 2 = 81.70 (approx. answer)

√89108 =?

√9,00,00 = 300

89108 ÷ 300 = 297.03

(297.03 + 300) ÷ 2 = 298.51 (approx. answer)

Another similar and easy way to approximate the square root of a number is to use the following equation:

The closer the known square is to the unknown, the more accurate the approximation. For instance, to estimate the square root of 15, we could start with the knowledge that the nearest perfect square is 16 (4²).

So we've estimated the square root of 15 to be 3.875. The actual square root of 15 is 3.872983...