How to Calculate Square Roots in Your Head

Learn how square roots are used in the real world, how you can easily estimate the value of a square root in your head, and how you can use an ancient algorithm to calculate the value of a square root by hand to as high a precision as needed.

Jason Marshall, PhD,
Episode #104
Square roots

How to Calculate Square Roots Without a Calculator


But what if that rough estimate just isn’t good enough and you need a more accurate answer? Are you out of luck? No, you can use a very cool and clever 2000+ year old algorithm known as the “Babylonian method” or “Heron’s method” to improve your estimate. Here’s how it works for the square root of 60:

  • Step 1 is to guess the answer…the better the guess the more quickly the accuracy of the estimate will improve. In our case, since we know the answer must be between 7 and 8, let’s guess 7.5. Since 7.5 x 7.5 = 56.25, we know this isn’t a super accurate guess, but let’s see what the algorithm can do for us.

  • Step 2 is to divide the number we’re taking the square root of by our guess. So 60 / 7.5 = 8.

  • Step 3 is to average this new number and our guess. In other words, add this number to our guess and divide the result by 2. That’s (7.5 + 8) / 2 = 7.75. This number is our new guess. Let’s check and see how good this guess is: 7.75 x 7.75 = 60.0625…which is really close to 60. So in only one trip through this sequence of steps we’ve improved our estimate of the square root of 60 tremendously.

  • Step 4 is to decide if your new guess is accurate enough. If it is, you’re done. If not, go back to the beginning with this new guess and repeat the process until the answer is as accurate as you need it to be. Every trip through this ancient algorithm will give you a more and more accurate answer.

Wrap Up

Okay, that’s all the math we have time for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new featured number or math puzzle posted every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at mathdude@quickanddirtytips.com.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!


About the Author

Jason Marshall, PhD

You May Also Like...

The Quick and Dirty Tips Privacy Notice has been updated to explain how we use cookies, which you accept by continuing to use this website. To withdraw your consent, see Your Choices.