What Is a Perfect Square? (2024)

By: Jesslyn Shields & Austin Henderson|Updated: Nov 14, 2023

What Is a Perfect Square? (1)

You know what a square is: It's a shape with four equal sides. Seems hard to improve upon, right? But what is a perfect square? In order to explain that, we'll have to dig into some math.



  1. Defining Perfect Squares
  2. Identifying Perfect Squares
  3. Characteristics of a Perfect Square
  4. The Beauty of Perfect Squares

Defining Perfect Squares

A perfect square is a special type of number that you get by multiplying another number by itself. It's like the mathematical equivalent of a square shape — neat, balanced, and intriguing.

To illustrate, when you multiply 5 by 5, you get 25, making 25 a perfect square because it's the result of 5 x 5.

Identifying Perfect Squares

So, how can you determine if a number is a perfect square? The trick lies in finding its square root, which is the reverse operation of squaring a number. If the square root yields a whole number, you're looking at a perfect square.

For instance, the square root of 16 is 4 because 4 x 4 = 16. Therefore, 16 is indeed a perfect square. Likewise, if you calculate the square root of 144, you'll find it's 12 because 12 x 12 = 144. Bingo! Another perfect square.


Perfect Squares Chart

To help you get a better grasp of perfect squares, here are a few more common examples:

  • 1 x 1 = 1
  • 2 x 2 = 4
  • 3 x 3 = 9
  • 4 x 4 = 16
  • 5 x 5 = 25
  • 6 x 6 = 36
  • 7 x 7 = 49
  • 8 x 8 = 64
  • 9 x 9 = 81
  • 10 x 10 = 100

As you can see, each of these numbers can be expressed as the product of two equal integers, making them perfect squares.


Characteristics of a Perfect Square

Now that we've defined perfect squares and how to identify them, let's explore a few more details to deepen our understanding.

When examining perfect squares, notice a fascinating pattern: The units place digit of a perfect square is always 0, 1, 4, 5, 6, or 9. This characteristic can be quite handy when identifying perfect squares quickly.


If you want to go a step further and find perfect squares manually, you can use long multiplication method.

Perfect squares are not limited to small numbers; they extend far beyond. For instance, 625 is a perfect square because it's the result of 25 x 25. Exploring larger perfect squares can be a fun mathematical journey.


The Beauty of Perfect Squares

Perfect squares are not just mathematical curiosities; they have real-world applications in various fields, including geometry and algebra. Understanding them can open doors to a deeper appreciation of mathematics.

Now That's Interesting

The Egyptians calculated square roots as far back as 1650 B.C.E.

This article was updated in conjunction with AI technology, then fact-checked and edited by a HowStuffWorks editor.


Frequently Answered Questions

Is there a perfect square formula?

The perfect square formula is simply squaring a whole number, such as "2," by multiplying it by itself: 2 x 2 = 4, and 4 is the perfect square.

How do I identify perfect squares?

To identify a perfect square, you can find its square root, which is the reverse operation of squaring a number. If the square root yields a whole number, you're looking at a perfect square.

Do perfect square numbers have any specific patterns in their digits?

The units place digit of a perfect square is always 0, 1, 4, 5, 6, or 9. This characteristic can be quite handy when identifying perfect square numbers quickly.

What is a perfect square trinomial?

A perfect square trinomial is an algebraic expression that can be factored into the square of a binomial. It often takes the form of (a + b)^2 or (a - b)^2, where "a" and "b" are integers.

What's the smallest whole number that is a square number?

The smallest whole number that is a square number is 1, as 1 x 1 = 1.

Is 12 a perfect square?

No, 12 is not a perfect square.

Is 72 a perfect square?

No, 72 is not a perfect square.

What is perfect square with an example?

A perfect square is a number that can be expressed as the product of two equal integers. For example, the number 36 can be expressed as the product of the integers 6 × 6.

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