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Perfect Squares – Definition, Formula, List, Examples

Jessica Kaminski

reviewed by Jo-ann Caballes

Updated on March 10, 2026

A perfect square is a number you get when you multiply a whole number by itself. For example, 25 is a perfect square because 5 x 5 = 25.

In this article, we’ll cover the definition of perfect squares, the formula to work them out, and perfect square numbers list charts, plus we’ll share practice questions so you can put your knowledge to the test.

What are the perfect squares?

The perfect square numbers are the numbers you get when you multiply an integer (aka a whole number) by itself. This is also known as squaring a number – so 5 x 5 is also 52 (squared), and the answer is 25.

Perfect square definition

The perfect square definition is the result of a number multiplied by itself. Since all of the perfect squares are a result of a number multiplied by itself, and we have infinite numbers, this means that there is an infinite number of perfect squares!

The reason these numbers are called perfect square numbers is that the final number can be arranged to form a perfect square shape made out of smaller squares – find out more about that below.

Perfect squares: Examples

There is an infinite number of perfect square examples, so making a full list is impossible. But below, you can find a list of perfect square roots of the first 20 numbers that are perfect squares.

You’ll see in our graphic below that when each digit of the result is represented by a square, they can all be formed to create an equal-sided square – hence the name perfect squares!

You might also notice a pattern here, in that every perfect square above only ends in the digits 0, 1, 4, 5, 6, and 9. This is true for all perfect squares, even beyond 400.

Perfect square formula

If our final number is ‘n’ and our squared number is ‘x’, the formula is: n = x2  or n x n. 

That’s because all you need to do to find a perfect square is multiply an integer by itself.

Perfect squares list

As there is an infinite number of perfect squares available, we cannot list them all here! Making a perfect square chart is also not possible for the same reason. Below, we’ve included the first 20 in the list of perfect squares:

• 1
• 4
• 9
• 16
• 25
• 36
• 49
• 64
• 81
• 100
• 121
• 144
• 169
• 196
• 225
• 256
• 289
• 324
• 361
• 400

It can also be helpful to visualize your perfect squares list as a chart, like the chart below. This helps kids visualize the effect of squaring numbers and understand the pattern that emerges with perfect squares.

Perfect square roots list and chart

How to identify perfect squares?

If you want to identify whether a number is a perfect square or not, you need to find its square root. If the square root is an integer (i.e., a whole number), you’ve identified a square number. 

For example, if we take 64, its square root is 8. Because 8 is a whole number, 64 is a perfect square. 66, however, has a square root of 8.12403840464. This number is not an integer, so 66 is not a perfect square.

Identifying a perfect square: Key observations

• Using perfect square factors
• Using patterns and rules
• All perfect squares are positive

Using perfect square factors

You can work out if a number is a perfect square by using its factors. With any given number, work out all of its integer factors. So if we take 100 as a perfect square example, its factors are:

• 1 (12 = 1)
• 2 (22 = 4)
• 4 (42 = 16)
• 5 (52 = 25)
• 10 (102 = 100)

Then, you can multiply all those numbers by themselves to see if any result in your number. You’ll see here that 10 x 10 = 100, so 100 is a square number.

If we take 99 as an example, its integer factors are:

• 1 (12 = 1)
• 3 (32 = 9)
• 9 (92 = 81)
• 11 (112 = 121)
• 33 (332 = 1089)

None of those numbers multiplied by themselves results in 99, so we can conclude that 99 is not a perfect square.

Using patterns and rules

All perfect squares end in one of the following digits: 0, 1, 4, 5, 6, 9. If your number doesn’t end in any of these digits, you can immediately eliminate it, knowing it is not a perfect square. In other words, any number ending in 2, 3, 7, or 8 is never a perfect square.

Check out these additional rules for perfect squares, too:

• If a number ends in an odd number of 0s, it’s never a perfect square – so 10 and 1,000 aren’t perfect squares (odd number of 0s), but 100 is (even number of 0s).
• Not all numbers ending with an even number of 0s are a perfect square – 100 and 400 are, but 300 is not a perfect square.

All perfect squares are positive

Our final trick is knowing that all perfect squares are positive. That’s because a positive multiplied by a positive is a positive, and a negative multiplied by a negative is a positive. All perfect squares are non-negative, because squaring any integer (positive or negative) gives a non-negative result.

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Perfecting the square

Now, let’s look at the expression “perfecting the square”, as it can raise a bit of confusion.

In our case, as we discussed, perfecting the square means finding the number that, when squared, yields a perfect square. In simple terms, you are checking whether a number can be written as a whole number times itself.

For example:

• 6 × 6 = 36

Since 36 is the result of multiplying 6 by itself, 36 is a perfect square.

Perfecting the square, however, also has another meaning in math, one you will encounter in a bit more advanced classes. It is a method we use to turn a quadratic expression into a perfect square trinomial. This strategy is for solving quadratic equations step by step, and also helps us better understand how perfect squares work in algebra.

Solved math tasks: examples

Which is a perfect square?

Out of these numbers, which is a perfect square?

• 12
• 35
• 52
• 81

Answer:

Find all of the perfect square factors of the above numbers:

• 12: 1, 2, 3, 4, 6
• 35: 1, 5, 7
• 52: 1, 2, 4, 13, 26
• 81: 1, 3, 9, 27, 81

Multiply each of those factors by themselves. The only factor that results in one of your four numbers is 9 into 81, so the answer is 81.

Work out the square root of 64

Work out the square root of 64, then tell us if it is a perfect square.

Answer:

The easiest way to work this out is to list all of the factors of 64, then multiply them by themselves:

• 1 x 1 = 1
• 2 x 2 = 4
• 4 x 4 = 16
• 8 x 8 = 64
• 16 x 16 = 236
• 32 x 32 = 1,024

Here, we can see that 8 x 8 = 64, making it a perfect square.

Find the perfect squares between 30 and 40.

Identify any and all perfect squares between 30 and 40.

Answer:

You can calculate this in a number of ways, from using your perfect squares lists to multiplying numbers from 1 upwards until you get an answer that’s larger than 40. Any number(s) sitting between 30 and 40 will be your answer.

Perfect squares: practice math problems

Perfect squares worksheet

Now that you’ve learnt all about perfect squares, including their definition, formula, and how to work them out from your end number, why not put your knowledge to the test with our engaging math worksheets?

• Systems of equations worksheets
• Factor tree worksheets
• Factoring trinomials worksheets
• 6th grade multiplication worksheets

Frequently asked questions on perfect squares

Is 25 a perfect square?

Yes, number 25 is a perfect square. It’s the perfect square of number five, meaning when you multiply 5 by 5 (or do 5 squared), you get 25. 

What are all the perfect squares?

Since the number of numbers is infinite, and each number has its perfect square, the number of perfect squares is infinite too! So, we can’t make a list of all perfect square roots. 

We’ve covered perfect squares up to 20 earlier in the article and included a convenient image chart.

How to find perfect squares?

To find the perfect square of each number, you can use two methods:

• Multiply it by itself, e.g., 6 x 6 = 36
• Raise it to the power of two, e.g., 7^2 = 49

These two operations are essentially the same, as they simply multiply the number by itself.

How to factor perfect squares?

To factor a perfect square, you need to find the number that multiplies by itself to give the original number. For example, since 8 × 8 = 64, the square root of 64 is 8. You can write 64 as 8² or 8 × 8. 

Jessica Kaminski

91 articles

Jessica is a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.