What is a perfect square? How do you know if a number is a perfect square?
In this article, we’ll define perfect squares, provide a list of the first 25 perfect squares (and the integers that make them up), and teach you how to tell if a number is a perfect square.
What Is a Perfect Square?
A perfect square is a number that can be expressed as the product of two equal integers.
What does that mean?
Basically, a perfect square is what you get when you multiply two equal integers by each other.
For instance:
$$5 * 5 = 25$$
25 is a perfect square because you’re multiplying two equal integers (5 and 5) by each other.
You can also express $5 * 5$ as $5^2$. That’s where you get the term “perfect square”
List of Perfect Squares
Here’s a list of the first 25 perfect squares. A hint: if you want to create a perfect square, simply square an integer!
| Square | Integers |
| 1 | $1 * 1$ |
| 4 | $2 * 2$ |
| 9 | $3 * 3$ |
| 16 | $4 * 4$ |
| 25 | $5 * 5$ |
| 36 | $6 * 6$ |
| 49 | $7 * 7$ |
| 64 | $8 * 8$ |
| 81 | $9 * 9$ |
| 100 | $10 * 10$ |
| 121 | $11 * 11$ |
| 144 | $12 * 12$ |
| 169 | $13 * 13$ |
| 196 | $14 * 14$ |
| 225 | $15 * 15$ |
| 256 | $16 * 16$ |
| 289 | $17 * 17$ |
| 324 | $18 * 18$ |
| 361 | $19 * 19$ |
| 400 | $20 * 20$ |
| 441 | $21 * 21$ |
| 484 | $22 * 22$ |
| 529 | $23 * 23$ |
| 576 | $24 * 24$ |
| 625 | $25 * 25$ |
How to Tell If a Number Is a Perfect Square
You can tell if a number is a perfect square in a couple of different ways.
First of all, if you create a square by multiplying two equal integers by each other, then the product is a perfect square.
So, $1 * 1$ is a perfect square. So is $10 * 10$ and $1,000 * 1,000$.
You can also tell if a number is a perfect square by finding its square roots. Finding the square root is the inverse (opposite) of squaring a number.
If you find the square root of a number and it’s a whole integer, that tells you that the number is a perfect square.
For instance, the square root of 25 is 5. The square root of 26 is not a whole integer. So, 26 is not a perfect square.
Key Takeaways: Understanding Perfect Squares
A perfect square is a number that can be expressed as the product of two equal integers.
You can tell if a number by finding its square root and seeing if that square root is a whole integer.
What's Next?
Getting ready to take the ACT? We have a list of 31 formulas you must know to conquer the ACT.
Taking the SAT instead? Here’s a list of our favorite SAT Math prep books that will help set you on the path to success.
Looking to brush up on your fundamental algebra skills? A good place to start is mastering systems of equations.
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Hayley Milliman is a former teacher turned writer who blogs about education, history, and technology. When she was a teacher, Hayley's students regularly scored in the 99th percentile thanks to her passion for making topics digestible and accessible. In addition to her work for PrepScholar, Hayley is the author of Museum Hack's Guide to History's Fiercest Females.
Holly R.