Estimating Square Roots

Watch this video to learn how to estimate square roots:

How to Estimate Square Roots: Video

Approximating Square Roots

What's the square root of 20 (√20)?

Since 20 is not a perfect square, we can find its approximate value by determining what 2 perfect squares it is between. We use the perfect squares as benchmarks.

a. You know that 20 is between 16 and 25.

b. Therefore √20 is between √16 and √25.

c. So, √20 is between 4 and 5. It could be 4.4 or it could be 4.7 but we know it's between 4 and 5.

Example

Find an approximate value for √39.

Solution

39 is between the perfect squares of 36 and 49.

√39 is between √36 and √49.

So √39 is between 6 and 7.

And it's closer to 6 than to 7 because 39 is closer to 36 than 49.

On a number line it could be represented as:

Can you tell by looking at the number line which other square roots would come between 6 and 7? (√40, √42, etc.)

Once you have an approximate value you can use guess and test to be more accurate.

Since we know the number is between 6 and 7 and is closer to 6 let's try 6.2.

6.2 x 6.2 = 38.44

6.3 x 6.3 = 39.69

We know the answer is between 6.2 and 6.3 but slightly closer to 6.2

6.25 x 6.25 = 39.0625

6.24 x 6.24 = 38.9376

We know the answer is between 6.24 and 6.25 but slightly closer to 6.25.

And we could continue to narrow the answer.

Estimating square root using a number line

It's possible to estimate square roots with some accuracy (to at least one decimal) by using a number line. For example, the square root of 13 would fall somewhere between the square roots of the perfect squares 9 and 16.

Between the square root of 9 and the square root of 16 mark off the square roots of 10-15.

Through estimating with a number line you can see that √13 will be approximately 3.6

Calculating Square Roots

A more accurate way of finding square roots of non-perfect numbers is to use your calculator. Many calculators have a square root button button that has the square root sign on it (BTW that sign - √ - is called a radical!) or it might just have 'sqrt' on it.

To find the square root, enter the number, then press the square root button. On some calculators you might press the square root button first, then the number, then enter. Your calculator will probably give you the answer to 10 or maybe more decimals.

√ 8 = 2.824271247461900976033774484194

Even this number is not accurate, it's only an approximation. The square root of 8 will go on forever and will never repeat. These square roots are often rounded to one or 2 decimal places. Rounded, the square root of 8 would be 2.83.

Examples

Example 1

A square has an area of 30cm². What is the length of one side?

Solution

To find the length of one side you'll need to determine the square root of 30.

The two closest perfect squares are 25 and 36 so it comes between the square roots of those numbers.

Draw a number line and mark off the square roots.

By marking off the positions on the number line you can see that √30 is between 5.4 and 5.5, just a bit closer to 5.5 if it was rounded off.

You can also use your calculator to find the square root of 30.

The side length of the square is approximately 5.48 cm.

Example 2

Between which two whole numbers is the square root of 80 (√80)?

Solution

The two closest perfect squares to 80 are 64 and 81 so √80 lies between √64 and √81.

√64 = 8 √81 = 9

Therefore √80 lies between 8 and 9

Example 3

5.4 is between the square roots of what two perfect squares?

Solution

5.4 is between the whole numbers 5 and 6.

5 is the square root 25 which is a perfect square. 6 is the square root of 36 which is a perfect square.

So 5.4 is between √25 and √36.