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.
