 # Squares & Perfect Squares – Explanation & Examples In mathematics, a square is a product of a whole number with itself. For instance, the product of a number 2 by itself is 4. In this case, 4 is termed as a perfect square.

A square of a number is denoted as n × n. Similarly, the exponential notation of the square of a number is n 2, usually pronounced as “n” squared. Square numbers are usually non-negative. ## What is a Perfect Square?

A perfect square is a number that is generated by multiplying two equal integers by each other. For example, the number 9 is a perfect square because it can be expressed as a product of two equal integers: 9 = 3 x 3. The first 25 perfect squares can be generated as shown in the table below:

Example 1

 Integer Perfect square 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 11 x 11 121 12 x 12 144 13 x 13 169 14 x 14 196 15 x 15 225 16 x 16 256 17 x 17 289 18 x 18 324 19 x 19 361 20 x 20 400 21 x 21 441 22 x 22 484 23 x 23 529 24 x 24 576 25 x 25 625

## How can you tell if a number is a Perfect Square?

There are many ways of determining if a number is perfect. A given number can be checked whether it is a perfect square using repeated division by prime factors.

Example 2

For example, to check whether 441 is a perfect square:

• Start by factorizing a number.

441 = 3 × 3 × 7 × 7

• Both numbers exist twice. Make two sets.
441 = 3 × 7 × 3 × 7
• Multiply them.

= 21 × 21

• It can be written as

= 212

• Hence, 441 is a perfect square.

You can also check whether a given number is a perfect square by finding the number’s square root. If the square root of a number is a whole number, then the number is a perfect square

For instance, the square root of 16 is 4. The square root of a number like 24 is not a whole number. Therefore, 24 is not a perfect square.

### Practice Questions

1. The area of a square coffee table is $3600$ cm². Which of the following shows the length of one side of the coffee table?

2. What is the length of a square flower garden if the area is 81 square yards.

3. The length of a square piece of paper is 4 cm. What is the area of the paper?

4. A square room has an area of 64 square feet. What is the length of a carpet needed to cover the floor of this room perfectly?

5. A square football pitch has a perimeter of $400$ meters. What is the area of this pitch?

6. The perimeter wall of a square compound is $480$ meters. What is the length and area of the compound?

7. The area of a square plot of land is $10000$ m². What is the perimeter of the plot of land?

8. A farmer divided his ranch into $5$ equal square sections. Suppose the length of each section is $50$ meters. What is the total area of the ranch?