Estimating Square Roots: Tips & Tricks

Published on Thursday, July 12, 2012 in , , , ,

purzen's (openclipart.org) thought bubble surrounding David Vignoni's square root icon over Josh Green's (subtlepatterns.com) Old Mathematics patternIn the previous post, you learned how to work out a close estimate of the square root of any number up to 1,000.

Building on that skill, this post will show you how to go a little farther with this. You'll learn some extra touches, how to display this skill in context, and more!


Sometimes, after giving your mental estimate, you recognize that the fraction can be simplified. In the first example from the previous post, the square root of 149 gave us an estimate of 12 and 5/25ths. It shouldn't be too hard to see that you can also say something like, “The square root is roughly 12 and 5/25ths, or 12 and 1/5th.”

If you're lucky enough to have a simplified fraction whose denominator is anywhere from 2 and 11, you can even give the decimal equivalent by using the approach taught in my Mental Division With Decimal Precision post.

For example, let's say you need to calculate the square root of 314. Your mental estimate should work out to be 17 and 25/35ths, simplified to 17 and 5/7ths. Using the decimal precision approach, you can add that the calculator will display this as roughly 17.7142857. You won't always be able to use this touch, but when you can use it, it's very impressive.

Estimating Error

Another handy trick is to estimate how close the square will be, before they square the number on their calculator to check your work. This is surprisingly easy to do. Once you've made your estimate and had the other person put it into the calculator, take a look at the total, and focus only the decimal point and the numbers to the right of it. In 28.38596, for example, you'd focus only on the .38596 part.

First, ask yourself if the decimal part of the number is less than .11111 (1/9th) or greater than .88888 (8/9ths). If so, the squared number will end in .9 something.

If that wasn't the case, ask yourself if the number is more .2857 (2/7ths) AND less than .7142 (5/7ths). If so, the square number will end in .7 something.

If the decimal portion of the number fails both of these tests, it will end in .8 something.

For example, let's say you're asked for the square root of 460, and you give the estimate of 21 and 19/43rds. When entered into the calculator, it appears as 21.44186 (approximately), and you focus on the .44186. This number isn't less than .11111 and isn't more than .88888, so the square won't end in .9. It is, however, between .2857 and .7142, so we know that the square will actually end in .7 something.

Since we know our estimates are always just under the given number, we can state that the square will be 459.7, and that it can't be less than .75. Before the other person squares the number on the calculator, you can say something like, “When you square that number, you won't get 460. Instead, you'll get 459.7 or so, but that's still pretty close.” They'll be impressed that not only can you estimate a square root, you can even estimate your margin of error!

Presenting in Context

It's best to present your square root ability in context, so I suggest having your audience help you make up a problem that will require you to demonstrate your square root ability.

The simplest example would be to start with a square land area, in square feet or meters, and work out a single side. Ask, “Imagine I have a square plot of land. How many square feet do I own? Give me any number from 1 to 1,000.” Once they give a number, further explain, “Now, the east side of my land sits right on the property line, so I need to build a fence the length of one side. So that fence would be roughly...” The length of the fence, of course, would the square root of the given land area.

Another good way to present your square root ability is with problems involving triangles and the Pythagorean theorem. Here's a quick 60-second refresher on the Pythagorean theorem for you:

What is the square root of 13? Using the tips you've already learned, you can state it's about 3 and 4/7ths, which the calculator will show as 3.571485, and when squared, will actually be close to 12.75 (the calculator shows 12.7551, roughly).

One simple, but fun, Pythagorean problem involves a cat stuck up on a pole. Ask your audience to imagine a cat stuck up so high on a pole that you can't reach it. It's anywhere from 8 to 20 feet high, so you ask for a number from 8 to 20.

You then mention that there's another obstacle. The ground around the pole, in a perfect circle, is too soft to support your ladder. Ask your audience what's the closest you can get to the pole while still being on solid ground, and that the answer should be any whole number from 1 to 20 feet.

The height of the cat on the pole can be thought of as a, and the closest your ladder can get to the pole can be thought of as b. Limiting both answers to no more than 20 feet will ensure that you'll never have to deal the square root of numbers over 1,000.

To determine how long the ladder is, you can use either of the techniques from my Squaring 2-Digit Numbers Mentally tutorial to square both sides, and add them together, and then get the square root of that total.

To keep the audience engaged, this should be done verbally. As an example, you might say, “You stated that the cat was stuck on a 17-foot pole, and the closest we could get with the ladder is 6 feet from the pole. Using the Pythagorean theorem, that's 289 plus 36, which is 325. The ladder needs to be the square root of 325, so the ladder should be roughly 18 and 1/37th feet long.”

Using the technique from the Estimating Error section above, you could also continue, “That's only a rough estimate, of course, because 18 and 1/37th, when squared, is actually about 324.9 something.”

If you want to find more stories and contexts in which to present your ability to do square roots, search the web for word problems involving square roots.

I hope you've found these lessons on estimating square roots to be useful and enjoyable. If you have any questions or comments, post them in the comments.

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