Square Root Calculator

Information about Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the symbol \sqrt{}.

For example, the square root of 9 is 3 because 3 times 3 equals 9. The square root of a number can be either positive or negative. For example, the square root of 9 is 3 or -3 because both 3 times 3 and -3 times -3 equal 9.

\sqrt{9} = 3

How to Find the Square Root of a Number Without a Calculator?

1. Start by guessing two numbers that you know the square roots of, and your number should be somewhere between them.
2. Take your number and divide it by one of the guesses you made in step 1.
3. Take the number you got from step 2 and the guess you used, then find the middle point between them.
4. Use the middle point to go back to step 2 and repeat the process until you get a number close enough to the answer you're looking for.

Let's Find the Square Root of 20

Let's figure out the square root of 20, but we'll only need to be close enough to get two decimal places right.

Step 1: we need to find two numbers whose square roots we know, and 20 should be between them. We know that: 

4 × 4 = 16

5 × 5 = 25

So, the square root of 20 is between 4 and 5.

Step 2: Let's divide 20 by the closer number, which is 4:

20 ÷ 4 = 5

Step 3: Now, let's find the middle point between 5 and 4. Add them together and divide by 2:

(5 + 4) ÷ 2 = 4.5

So, our new guess is 4.5.

Step 4: Let's repeat the process with 4.5:

First, divide 20 by 4.5: 

20 ÷ 4.5 ≈ 4.44

Now, find the middle point between 4.44 and 4.5: 

(4.44 + 4.5) ÷ 2 ≈ 4.47

So now, we guess 4.47

Step 5: Let's see if 4.47 is close enough:

Multiply 4.47 by 4.47: 

4.47 × 4.47 ≈ 19.98

This is really close to 20, so the square root of 20 is approximately 4.47.

If you're happy with this answer, you can stop here! If you want to be even more precise, you could repeat the steps again.

Some common list of perfect square roots

√1 = 1, since 1 × 1 = 1

√4 = 2, since 2 × 2 = 4

√16 = 4, since 4 × 4 = 16

√25 = 5, since 5 × 5 = 25

√36 = 6, since 6 × 6 = 36

√49 = 7, since 7 × 7 = 49

√64 = 8, since 8 × 8 = 64

√81 = 9, since 9 × 9 = 81

√100 = 10, since 10 × 10 = 100

√121 = 11, since 11 × 11 = 121

√144 = 12, since 12 × 12 = 144

√169 = 13, since 13 × 13 = 169

√196 = 14, since 14 × 14 = 196

√225 = 15, since 15 × 15 = 225

√256 = 16, since 16 × 16 = 256

√289 = 17, since 17 × 17 = 289

√324 = 18, since 18 × 18 = 324

√361 = 19, since 19 × 19 = 361

√400 = 20, since 20 × 20 = 400

List of imperfect square roots

√2 ≈ 1.41421356237

√3 ≈ 1.73205080757

√5 ≈ 2.2360679775

√6 ≈ 2.44948974278

√7 ≈ 2.64575131106

√8 ≈ 2.82842712475

√10 ≈ 3.16227766017

√11 ≈ 3.31662479036

√12 ≈ 3.46410161514

√13 ≈ 3.60555127546

√14 ≈ 3.74165738677

√15 ≈ 3.87298334621

√17 ≈ 4.12310562562

√18 ≈ 4.24264068712

√19 ≈ 4.35889894354

√20 ≈ 4.47213595499

√21 ≈ 4.58257569496

√22 ≈ 4.69041575982

√23 ≈ 4.79583152331

√24 ≈ 4.89897948557

√26 ≈ 5.09901951359

√50 ≈ 7.07106781187

√80 ≈ 8.94427190999

√90 ≈ 9.48683298051

√99 ≈ 9.94987437107

√101 ≈ 10.0498756211