Perfect Square

Perfect Square is a number obtained by multiplying a whole number by itself, like 4 which is obtained when 2 is multiplied by itself, i.e. 2 × 2 = 4, thus 4 is a perfect square. In mathematical terms, the perfect square is expressed as a².

In this article, we have covered the meaning and definition of perfect squares, methods of finding perfect squares, and a list of perfect squares and applications.

What is Perfect Square?

Perfect squares are numbers that you get when you multiply a whole number by itself. For instance, 4 is a perfect square because it’s 2 times 2. Another example is 9, which is 3 times 3. These numbers have a special property, being the result of multiplying a whole number by itself. Examples of perfect squares include 1, 4, 9, 16, and so on.

Definition of Perfect Squares

Perfect square is a number achieved by multiplying a whole number by itself. For example, 4 is a perfect square since it is the product of 2 multiplied by 2.

How to Identify Perfect Square Numbers?

To find a perfect square number, take a whole number and multiply it by itself. For example, let’s consider the number 16. If we take the whole number 4 and multiply it by itself (4 × 4), the result is 16.

Since the outcome is a whole number, 16 is a perfect square. In general, this method helps determine if a number is a perfect square by checking if it can be expressed as the product of a whole number multiplied by itself.

Perfect Square Formula

Formula for a perfect square is expressed as n², where ‘n‘ is a whole number. In this formula, n is multiplied by itself, resulting in a perfect square. For example, if n is 3, the perfect square is 3², which equals 9.

Other formulas used for perfect square are,

• n² − (n − 1)² = 2n − 1

• n² = (n − 1)² + (n − 1) + n

Algebraic Identities as perfect squares:

• a² + 2ab + b² = (a + b)²

• a² – 2ab + b² = (a – b)²

Perfect Squares from 1 to 100

List of perfect squares from 1 to 100 is added in the table below,

Perfect Square Numbers From 1 to 100
1	=	1 × 1	=	12
4	=	2 × 2	=	22
9	=	3 × 3	=	32
16	=	4 × 4	=	42
25	=	5 × 5	=	52
36	=	6 × 6	=	62
49	=	7 × 7	=	72
64	=	8 × 8	=	82
81	=	9 × 9	=	92
100	=	10 × 10	=	102

List of Perfect Squares from 1 to 100

List of perfect squares between 1 to 100 are shown in the table below:

12 = 1	112 = 121	212 = 441	312 = 961	412 = 1681	512 = 2601	612 = 3721	712 = 5041	812 = 6561	912 = 8281
22 = 4	122 = 144	222 = 482	322 = 1024	422 = 1764	522 = 2704	622 = 3844	722 = 5184	822 = 6724	922 = 8464
32 = 9	132 = 169	232 = 529	332 = 1089	432 = 1849	532 = 2809	632 = 3969	732 = 5329	832 = 6889	932 = 8649
44 = 16	142 = 196	242 = 576	342 = 1156	442 = 1936	542 = 2916	642 = 4096	742 = 5476	842 = 7056	942 = 8836
52 = 25	152 = 225	252 = 625	352 = 1225	452 = 2025	552 = 3025	652 = 4225	752 = 5625	852 = 7225	952 = 9025
62 = 36	162 = 256	262 = 676	362 = 1296	462 = 2116	562 = 3136	662 = 4356	762 = 5776	862 = 7396	962 = 9216
72 = 49	172 = 289	272 = 729	372 = 1369	472 = 2209	572 = 3249	672 = 4489	772 = 5929	872 = 7569	972 = 9409
82 = 64	182 = 324	282 = 784	382 = 1444	482 = 2304	582 = 3364	682 =4624	782 = 6084	882 = 7744	982 = 9604
92 = 81	192 = 361	292 = 841	392 = 1521	492 = 2401	592 =3481	692 =4761	792 = 6241	892 = 7921	992 = 9801
102 = 100	202 = 400	302 = 900	402 = 1600	502 = 2500	602 =3600	702 =4900	802 = 6400	902 = 8100	1002 = 10000

Properties of Perfect Squares

Some important properties of perfect square are,

Result of Squaring an Integer	Perfect square is result of multiplying an integer by itself.
Negative Numbers Can Form Perfect Squares	Negative integers can form perfect square, e.g., (−4)2 = 16
Unique Square for Each Integer	Each integer has not a unique square. Two integers have one square, i.e. ‘a’ and ‘-a’ have same square.
Zero is a Perfect Square	Zero is considered a perfect square because 02 = 0
Sum of Consecutive Odd Numbers	A perfect square is sum of consecutive odd numbers.
Geometric Representation	Perfect square represents area of any figure.

Perfect Square Chart

Chart for Perfect Square is added below as,

Perfect Square Tips and Tricks

Some Tricks and Tips for Perfect Squares are given below.

Square of a Number Ending in 5: To find square of a number ending in 5, multiply the digit before 5 with next digit and append 25. For example, 75²= 7×8(25) = 5625

Square of Numbers Close to 100: For numbers close to 100, express the square as (100 – x)²= 100² – 200x + x². This simplifies calculations, especially for mentally calculating squares.

Odd Number Squares: Square of any odd number is an odd number. If n is an odd number, then n² is odd.

Even Number Squares: Square of any even number is an even number. If m is an even number, then m² is even.

Difference of Squares: Use difference of squares formula, a²− b²= (a+b)(a−b). This can help in factoring or simplifying expressions.

Square of a Sum: (a+b)² = a² + 2ab + b²

Square of a Difference: (a−b)² = a² − 2ab + b²

Observations on Perfect Squares

Perfect numbers end with any one of these digits 0, 1, 4, 5, 6, or 9. Also some observations on perfect squares are,

• Numbers ending with 3 and 7 have 9 as units place digit in their square number.

• Numbers ending with 5 have 5as units place digit in their square number.

• Numbers ending with 4 and 6 will have 6 as units place digit in their square number.

• Numbers ending with 2 and 8 will have 4 as units place digit in their square number.

• Numbers ending with 1 and 9 will have 1 as units place digit in their square number.

How many Perfect Squares are between 1 and 100?

There are 8 perfect squares between 1 and 100 (excluding 1 and 100). They are,

4, 9, 16, 25, 36, 49, 64 and 81

How many Perfect Squares are between 1 and 1000?

There are 30 perfect squares between 1 and 1000. They are,

4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900 and 961

Examples of Perfect Squares

Example 1: Identify the first two perfect squares.

Solution:

First two perfect squares are obtained by squaring the first two whole numbers:

• 1²=1 (Square of 1 is 1)
• 2²= 4² (Square of 2 is 4)

Therefore, first two perfect squares are 1 and 4.

Example 2: If a number is a perfect square and its square root is 9, what is the number?

Solution:

If a number is a perfect square and its square root is 9, we can find the number by squaring the square root:

9² = 81

So, required number is 81, as it is a perfect square, and its square root is 9.

Example 3: If a number is a perfect square and its square root is a prime number, find the number.

Take the prime number 5. The square of 5 is 25 (5²=25). Here, 25 is a perfect square, and 5 is a prime number.

So, the number we’re looking for is 25, where the square root (5) is a prime number

Practice Questions on Perfect Squares

Some questions on perfect square are,

Q1: Find the square of 5.

Q2: Is 36 a perfect square?

Q3:. Determine the square root of 49.

Q4: Write next two perfect squares after 16.

Q5: Identify the perfect square closest to 150.

Frequently Asked Questions on Perfect Squares

How many Perfect Squares are between 1 and 100?

There are 10 perfect squares between 1 and 100. These are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

How many Perfect Squares are between 1 and 1000?

There are 31 perfect squares between 1 and 1000. These include numbers like 1, 4, 9, 16, 25, and so on, up to 961.

Is 216 a perfect square?

Yes, 216 is a perfect square. The square root of 216 is 14, because 14 multiplied by itself (14 × 14) equals 216.

What defines a perfect square?

A perfect square is a number that can be made by multiplying a whole number by itself. For instance, 9 is a perfect square because it’s 3 times 3.

How does one determine if a number qualifies as a perfect square?

To check if a number is a perfect square, you see if it can be expressed as the product of a whole number multiplied by itself. If yes, it’s a perfect square.

In mathematical terms, what characterizes a perfect square trinomial?

A perfect square trinomial in math is an expression that can be factored into two identical binomials. It has the form (a+b)².

Which numerical values are considered perfect squares?

Numbers like 1, 4, 9, 16, and so on, are perfect squares. They result from multiplying a whole number by itself.

What is the process for factoring perfect squares?

To factor perfect squares, you write them as the square of a binomial. For example, 25=(5)²

What approach is used to identify perfect squares?

Identifying perfect squares involves finding if a number can be written as the product of a whole number multiplied by itself.

Does the number 7 qualify as a perfect square?

No, 7 is not a perfect square. You can’t get it by multiplying a whole number by itself.