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

IntegerPerfect square
1 x 11
2 x 24
3 x 39
4 x 416
5 x 525
6 x 636
7 x 749
8 x 864
9 x 981
10 x 10100
11 x 11121
12 x 12144
13 x 13169
14 x 14196
15 x 15225
16 x 16256
17 x 17289
18 x 18324
19 x 19361
20 x 20400
21 x 21441
22 x 22484
23 x 23529
24 x 24576
25 x 25625

 

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 cm 2. Calculate the length of one side of the coffee table.
  2. Calculate 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 square feet. Calculate 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. Calculate 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 m 2. Find 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. Calculate the total area of the ranch.

 

Solutions to practice questions

1. The length of the coffee table is found by determining the square of the area:

3600 = 60 x 60

The length of the table is 60 cm.

2. 625 = 25 x 25

The length of the flower garden is 25 yards.

3. Area of piece = 4 x 4

= 16 cm 2.

4. 81 = 9 x 9

The length of the carpet is 9 feet

5. Length of pitch is calculated as: 400 ÷ 4 =100 meters

Area of pitch = 100 x 100 =10000 square meters.

6. Length of compound = 480 ÷ 4 = 120 meters

Area of compound = 120 x 120 = 14400 square meters.

7. 10000 = 100 x 100,

Therefore, perimeter of the square plot is calculated as:

100 x 4 = 400 meters

8. Area of each piece is 50 x 50 = 2500 square meters.

The total area of ranch is 2500 x 5 = 12500 square meters

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