What Is a Perfect Square in Math? A Complete Guide for Middle Schoolers [Part 2]

This easy-to-follow guide to perfect squares in math is here to help you out, whether you're just learning about them or need a quick reminder. You’ll discover easy explanations, helpful tips, and fun activities!

For a more complete overview of the topic, we recommend visiting the Part 1: What Does Squared Mean in Math? guide.

What Is a Perfect Square?  

A perfect square — or simply, a square — is a number that we get when we multiply a whole number by itself.

Let's see how that works in numbers:

When we multiply 3 by itself, which can be expressed with 3×3 or 3², we get 9.

So, 9 is a perfect square.

We can square any whole number, i.e. multiply it by itself, and get a perfect square:

• 4² = 16
• 5² = 25
• 6² = 36
• And so on

In these examples, the numbers 16, 25, and 36 are all perfect squares.

We can present perfect squares visually by treating each whole number that builds into it as a row of little blocks, like so:

Now let’s count the blocks:

• To square the number 2, we place 2 blocks into 2 rows, and we get 4 blocks in total. 4 is a perfect square.
• To square the number 3, we place 3 blocks into 3 rows, and we get 9 blocks in total. 9 is a perfect square.
• To square the number 4, we place 4 blocks into 4 rows, and we get 16 blocks in total. 16 is a perfect square.
• To square the number 5, we place 5 blocks into 5 rows, and we get 25 blocks in total. 25 is a perfect square.
• And so on.

As you can see, placing the blocks into an equal number of rows forms a perfect square. To get our square number, all we have to do is count the blocks.

List of Perfect Squares

Here are some more examples of perfect squares.

The Difference Between Perfect and Non-Perfect Squares

The easiest way to determine whether a number is a perfect square or not is by finding its square root. A square root is the number we have to multiply by itself to get a square number.

The square root of a perfect square is always a whole number.

Take the number 25.

The square root of 25 is 5 because 5x5 = 25.

Since 5 is a whole number, 25 is a perfect square.

Now, let's try the number 30.

What number do we have to multiply by itself to get 30? In other words: What is the square root of 30?

That number is approximately 5.477 (we read this as "five and four hundred seventy-seven thousandths"). Whenever you see the decimal point in a number, you will know that it is not a whole number.

So, the number 5.477 is not a whole number.

As we said, the root of a perfect square is a whole number, so we can conclude that the number 30 is not a perfect square.

Another useful clue for determining if a number is a perfect square or not is to look at its last digit.

Go back to our list of perfect squares and notice that none of the numbers end in 2, 3, 7, or 8.

Let’s test this on large numbers:

• Numbers 402, 412, 422, 432, etc., are not perfect squares. For instance, the square root of 402 is approximately 20.05 which is not a whole number.
• Let's try the numbers ending in 3, such as 403, 413, 423, 433, etc. For example, the square root of 413 is approximately 20.32, which is not a whole number and, therefore, 413 is not a perfect square.
• Let's try the numbers ending in 7, such as 407, 417, 427, 437, etc. For example, the square root of 407 is approximately 20.17, which is not a whole number and, therefore, 407 is not a perfect square.
• Lastly, looking at numbers like 408, 418, 428, etc., none of these are perfect squares either. The square root of 408 is approximately 20.2 which is not a whole number, so 408 is not a perfect square.

Although not a rule, one more pattern you can find by observing the last digits of perfect squares is that they might end in numbers like 1, 4, 5, 6, 9, and 00.

Examples of this include:

121 = 11 x 11

324 = 18 x 18

625 = 25 x 25

196 = 14 x 14

169 = 13 x 13

400 = 20 x 20

Check out our video for a full guide to finding perfect squares and square roots.

How to Square Numbers (with Our Tutors’ Tips)

By now, you know that the most common path to finding perfect squares is to multiply a whole number by itself. When we need to square a multi-digit number, we use the long multiplication method.

For instance, we can use the long multiplication method to square 23.

• Step 1: Write the number you want to square. Then, write it again below.
• Step 2: Start by multiplying the ones digit of the bottom number (3) by the top number (23).
• Step 3: Now, multiply the tens digit of the bottom number (2) by the top number (23), but start one place to the left and add a zero to the right of the result.
• Step 4: Finally, add up both of the partial products to get your final answer.
  
  

Pretty helpful, right?

However, this process gets more complicated with larger numbers. That’s why our tutors have put together a couple of tricks to help you find perfect squares more quickly:

1. Squaring Numbers That End in 5

Let’s choose a number ending in 5, like 35.

To find the perfect square of 35, we’ll follow these simple steps:

• Step 1: Separate 3 and 5.
• Step 2: Multiply 3 by its successor, number 4.   3 x 4 = 12
• Step 3: Square number 5.   5² = 25
• Step 4: Put the digits of the second step (12) and the third step (25) together. You get 1225.

2.  Using Perfect Square Trinomial Formula

This method might be interesting to high school students who have already mastered the trinomial formula. To refresh our memory, the trinomial formula looks like this:

(a + b)² = a² + 2ab + b²

This method breaks a large number into smaller, more manageable parts, and helps us figure out the answer without having to take too many steps.

Now, let’s illustrate this formula with the same number as in the previous section, 35.

We can express 35 as a sum of two numbers, 30 and 5. So in this case a = 30, b = 5 

35² = (30 + 5)² = 30² + 2 (30) (5) + 5² 

 = 900 + 300 + 25

= 1225

How We Use Perfect Squares Outside the Classroom

Perfect squares are not just limited to textbook pages and classroom walls. They are used in countless ways in our daily lives, especially when we are designing, constructing, and organizing spaces and objects.

Here are a few examples of perfect squares we use in everyday life.

Minecraft¹

Minecraft, the wildly popular sandbox game, is renowned for its limitless creative potential.

Within its pixelated landscapes, players can construct anything their imaginations can come up with, from towering castles to intricate redstone contraptions. At the heart of many of these creations lies the perfect square.

• Building a House: In Minecraft, when you want to make a house, you start by putting down square blocks in a line to make the bottom part.
• Making Walls and Rooms: After you have the bottom part, you add more blocks on top to make walls. Each wall is like a big square, and you can make different rooms by adding walls in different places.
• Adding Floors and Ceilings: When your walls are up, you can make floors and ceilings by adding more square blocks. This helps make each level of the house fit together nicely.

Rubik’s Cube

The Rubik's Cube is another example of perfect squares in action. It's made up of smaller squares arranged in a 3x3 grid on each face, challenging players to solve its colorful puzzle by twisting and turning its sides.

Sudoku

Sudoku puzzles are filled with perfect squares! The puzzle grid consists of nine rows and nine columns, forming nine smaller 3x3 squares within it. The goal is to fill each row, column, and 3x3 square with the numbers 1 through 9 without repeating any.

Solved Examples of Perfect Squares

 Using the perfect square trinomial formula, let’s find the square of 52. We can express 52 as a sum of two numbers, 50 and 2. So in this case a = 50, b = 2

The perfect square trinomial formula is: (a + b)² = a² + 2ab + b²

52² = (50 + 2)² = 50² + 2(50)(2) + 2²

52 = 2500 + 200 + 4

52² = 2704

2-Minute Quiz! How Well Do You Know Perfect Squares?

 1) Which of the following is NOT a perfect square?

A) 16

B) 25

C) 32

2) Which numbers are never perfect squares?

A) Numbers ending in 3, 6, or 9

B) Numbers ending in 2, 7, or 8

C) Numbers ending in 1, 4, or 5

3) The square root of 196 is ____.

• 13
• 16
• 14

4) What are the perfect squares between 1 and 10?

A) 1,4,6

B) 1,4,9

C) 1,6,9

Keep scrolling to find the quiz answers!

Perfect Squares FAQs

Find answers to common questions about the definition, properties, and examples of perfect squares.

1) Can negative numbers be perfect squares?

Perfect squares are never negative numbers. They're always positive whole numbers because they show the size of a square, and you can't have negative sizes.

 2) Is zero a perfect square?

When we multiply 0 by itself, we always get 0. Similarly, the square root of 0 is also 0. That's why we think of it as a perfect square.

3) Can we find the square root of a number that is not a perfect square?

Yes, but the square roots are not whole numbers. In fact, they are not even terminating or repeating decimals like 0.5, 0.75, or 0.6.

The square root of a non-perfect square is an irrational number.

The decimal representations of irrational numbers never terminate or repeat.

Example:

√2=1.41421356237309504880… 

4) What is a perfect square trinomial?

Perfect square trinomials aren’t usually part of the middle school math curriculum unless you are an advanced student. Nevertheless, it is a question related to perfect squares, so let’s explore it!

In algebra, our knowledge of perfect squares helps us to simplify expressions and solve equations by recognizing patterns like the trinomial formula we’ve seen earlier, which allows us to factor quadratic expressions and solve quadratic equations more efficiently.

A perfect square trinomial is an algebraic expression that comes from squaring a binomial. A binomial is the addition or subtraction of two terms, like this:

(x + 3).

To get a perfect square trinomial, you multiply the binomial by itself:

Step 1: Multiply the binomial by itself:

(x + 3)(x + 3)

Step 2: Multiply each term in the binomial:

= x * x + x * 3 + 3 * x + 3 * 3

= x² + 6x + 9

The final expression, x² + 6x + 9 , is a perfect square trinomial.

There are two common formulas for perfect square trinomials:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

Learn & Master Perfect Squares with Math Tutors Near You

Mathnasium’s specially trained tutors work with students of all skill levels to help them learn and master any K-12 math topic, including perfect squares.

Our tutors assess each student’s skills to develop personalized learning plans that will put them on the best path towards math excellence.

Find a Mathnasium Learning Center near you, schedule an assessment, and enroll today!

Find a Math Tutor Near You

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^{1. Not approved by or associated with Mojang or Microsoft.}

^{2. Quiz Answers:  1 - C, 2 - B, 3 - C, 4 - B.}