How do you find the square root of a number without a calculator?

Question

The Pythagorean Theorem

Answer 1

One of the ways is to use Newton's divide-and-average method.  Let's try the square root of 45.  We can only get the approximate value.

Step 1. 45 is between perfect squares 36 and 49. Since 45 is closer to 49 than to 36, choose 7, which is the square root of 49.

Step 2. 45 / 7 = 6.4285

Step 3. 6.4285 + 7 = 13.4285 (the 7 we added here is still the same 7 we chose from step 1)

Step 4. 13.4285 / 2 = 6.7143  (the divisor is 2 as we were getting the average of 7 and 6.4285).  Therefore the square root of 45 is approximately 6.7143.

If you want to get a closer approximation, repeat steps 1 and 2 using 6.7 as the divisor.