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Squares and Cubes: List of Squares and Cubes 1 to 100

Squares and Cubes are mathematical operations involving numbers that are essential in various areas of mathematics. Square of a number is obtained by multiplying the number by itself where as a cube of a number is obtained by multiplying the number by itself twice.

In this article, we will learn what is Square and Cube Number. We will also learn about Perfect Squares and Cube and Square and Cube charts 1 to 100.

squares-and-cuves

Table of Content

What is Square of a Number?

When an integer is multiplied by itself, it is called a square of that number. In simple words, a number that is multiplied two times is known as a square number. A square number is denoted as ‘n2 ‘ in mathematics.

Examples of Square Numbers:

Suppose a number ‘7’ is given. To find its square, just multiply it again by ‘7’. Here, we get 7⨯7= 49. So, ’49’ is the square of ‘7’. Some more examples of finding a square number are below:

  • 52 = 5⨯5= 25
  • 122 = 12⨯12= 144

What is Cube of a Number?

When we multiply an integer three times by itself, it is called a cube of that number. In other words, when an integer is multiplied by its square, it becomes a cube number. It is denoted as ‘n3 ‘ in mathematics.

Examples of Cube Numbers

Let us take an integer ‘3’. First find its square number 3⨯3= 9. Now, multiply ‘9’ with 3 again 9⨯3= 27. Here, ’27’ is called the cube of ‘3’.

Also, we can simply multiply it thrice to find its cube. Suppose a number ‘6’. Multiply it three times by itself 6⨯6⨯6= 216. The cube of ‘6’ is ‘216’. Some more example are as follows:

  • 23 = 8
  • 113 = 1331

Square and Cube 1 to 20

In this, we will learn the square and cube of numbers from 1 to 20. Let’s have a look on them

Square 1 to 20

Number Square Number Square
1 1 11 121
2 4 12 144
3 9 13 169
4 16 14 196
5 25 15 225
6 36 16 256
7 49 17 289
8 64 18 324
9 81 19 361
10 100 20 400

Cube 1 to 20

Number Cube Number Cube
1 1 11 1331
2 8 12 1728
3 27 13 2197
4 64 14 2744
5 125 15 3375
6 216 16 4096
7 343 17 4913
8 512 18 5832
9 729 19 6859
10 1000 20 8000

Squares and Cubes from 1 to 30

Squares and cubes of any number are very important for solving complex mathematical problems. They provide a basic idea to evaluate a question. Every student should memorize the squares and cubes from 1 to 30 as these serve as the basic pillars for simplifying problems.

Table of Squares and Cubes (1 to 30)

In this section, we will learn the square and cubes from 1 to 30. This will help students to solve the problems related to arithmetic operations. For any student, these are the basic squares and cubes which helps them to calculate easily and quickly. Here is the table in which the squares and cubes from 1 to 30 are given:

Squares 1 to 30

Square-1-to-30

Cubes 1 to 30

Cubes-1-to-30

Perfect Square and Cubes

Perfect Squares and Cubes are those numbers whose square root and cube root is a natural number or an intger in case of cube. Not every number we come across is a perfect square or cube. Hence, we need to learn what are perfect squares and cubes and also learn how to check a perfect square or cube.

Perfect Square

A perfect square is the square of an integer that can be multiplied by itself two times. For example ’16’ is a perfect square because 4⨯4= 16.

Perfect Cube

A perfect cube is the cube of an integer that can be multiplied by itself three times. For example, ’27’ is a perfect cube because 3⨯3⨯3= 27.

How to Identify Perfect Squares and Cubes?

After learning the definition of both perfect squares and perfect cube, we learn some easy ways for their identification. First, we learn about ‘unit digit’ or ‘end digit’ method, then we study another method prime factorization:

Unit Digit Method

Unit Digit Method is helpful in knowing about the possibility of a number being perfect square or cube without any actual test just by looking at the unit digit of a number. Let’s learn more about it.

  • Squares: If a number is a perfect square, its end digit should only be 0, 1, 4, 5, 6, or 9. Any square whose unit digit is any other digit cannot be a perfect square. For example, the square of ‘7’ is ’49’. Here, 49 is a perfect square because its unit digit is ‘9’. Similarly, take the square of ’12’. The square of ’12’ is ‘144’ which is a perfect square, because its unit digit is ‘4’.
  • Cubes: The unit digit of a perfect cube should only be 0, 1, 8, or 9. In this way, we can easily verify whether a number is a perfect cube or not. For example, ‘1728’ is a perfect cube because its last digit is ‘8’.

Note: There are also some exceptions. Some numbers are both a ‘perfect square’ and a ‘perfect cube’. For example, ’64’ is both perfect square and perfect cube.

Prime Factorization Method

Since, Unit digit method only gives a hint about the possibility of a number being perfect square or cube. However, the actual clarity can be gained only through prime factorization method.

  • Squares: In this method, when we prime factorize any number, each group of prime factors should have an even exponent i.e. ‘multiples of 2′. For example, the given number is ’36’. Its prime factor is 22 ⨯ 32 . Each group of prime factors has an even exponent ‘2’ therefore 36 is a perfect square.
  • Cubes: In this method, when we prime factorize any number, each group of prime factors should have an exponent that is multiple of ‘3’. For example, the given number is ’64’. Its prime factor is 26 . The exponent ‘6’ is multiple of ‘3’ therefore 64 is a perfect cube.

Squares and Cubes from 1 to 50

Here, a list of squares and cubes from 1 to 50 is given. Learning these values will help students to reduce their calculation time and they can easily solve complex problems.

Squares 1 to 50

Number Square Number Square Number Square Number Square Number Square
1 1 11 121 21 441 31 961 41 1681
2 4 12 144 22 484 32 1024 42 1764
3 9 13 169 23 529 33 1089 43 1849
4 16 14 196 24 576 34 1156 44 1936
5 25 15 225 25 625 35 1225 45 2025
6 36 16 256 26 676 36 1296 46 2116
7 49 17 289 27 729 37 1369 47 2209
8 64 18 324 28 784 38 1444 48 2304
9 81 19 361 29 841 39 1521 49 2401
10 100 20 400 30 900 40 1600 50 2500

Cube 1 to 50

Number Cube Number Cube Number Cube Number Cube Number Cube
1 1 11 1331 21 9261 31 29791 41 68921
2 8 12 1728 22 10648 32 32768 42 74088
3 27 13 2197 23 12167 33 35937 43 79507
4 64 14 2744 24 13824 34 39304 44 85184
5 125 15 3375 25 15625 35 42875 45 91125
6 216 16 4096 26 17576 36 46656 46 97336
7 343 17 4913 27 19683 37 50653 47 103823
8 512 18 5832 28 21952 38 54872 48 110592
9 729 19 6859 29 24389 39 59319 49 117649
10 1000 20 8000 30 27000 40 64000 50 125000

Patterns in Squares and Cubes

There are some interesting patterns in squares and cubes which often show some distinct properties and mathematical relations. Here are some important patterns which every student should know:

Patterns in Square Numbers

There are various patterns in the square numbers, some of those patterns are:

  • Difference between square numbers

The difference between any two consecutive squares is always an odd number. For example, Two consecutive squares ‘4’ and ‘9’ are given. Their difference is 9-4= 5, which is an odd number.

  • Sum of consecutive natural numbers

Whenever we square any odd number, the resultant will always be the sum of two consecutive natural numbers. Suppose we take the square of ‘3’ which is ‘9’. Here, ‘9’ is the result of the addition of two consecutive numbers ‘4’ and ‘5’.

  • Product of two consecutive even or odd natural numbers

The product of two consecutive even numbers or consecutive odd numbers is also an important pattern of square numbers. For example, ’25’ is the product of odd numbers 5⨯5. Similarly, ’64’ is the product of two even numbers 8⨯8.

  • Adding first n odd numbers

A square of any number is obtained by the sum of first ‘n’ odd numbers. Suppose, a number is given ’25’. Here, 25 is obtained by the addition of the first 5 odd numbers i.e. (1+3+5+7+9).

  • Adding triangular numbers

Triangular numbers are the numbers obtained by adding the next natural number and it forms an equilateral triangle. The formula to find triangular numbers is:

T(n)= 1+2+3+4…….+n

Now, by adding these triangular numbers, square numbers can be generated easily. For example, the square number is ‘4’, which is the addition of the first triangular number to itself i.e. (1+3).

Patterns in Cube Numbers

Some commons patterns in cubes are:

  • Adding consecutive odd numbers

By adding consecutive odd numbers, we can easily find the next cube numbers. For example, the cube of ‘1’ is 1. Now, add the next pair of consecutive odd numbers to find the next cube. Here, 3+5= 8 which is the cube of ‘2’. Similarly, to find the cube of ‘3’, add the next set of consecutive odd numbers 7+9+11= 27.

  • Difference of Cubes of Two consecutive positive integers

The difference between the two consecutive positive integers can give a cubic number. For example, The difference between 23 – 13 = 7. This represents 23 = 8. Similarly, the difference between 33 – 23 = 19 which represents 33 = 27.

  • Triangular Number Pattern

Similar to square numbers, we can also find cubic numbers by adding the triangular numbers. For example, the cube number is ‘23‘, which is the addition of the next triangular numbers i.e. (1+2=3).

Chart of Squares and Cubes 1 to 100

This chart will help students to learn the squares and cubes from 1 to 100. It will help them to solve problems of various mathematical topics such as algebra, geometry and arithmetic. Also squares and cubes are part of the number theory that will help them to deeply understand the integers.

Squares from 1 to 100

The following table shows the squares from 1 to 100:

Number Square Number Square Number Square Number Square Number Square
1 1 21 441 41 1681 61 3721 81 6561
2 4 22 484 42 1764 62 3844 82 6724
3 9 23 529 43 1849 63 3969 83 6889
4 16 24 576 44 1936 64 4096 84 7056
5 25 25 625 45 2025 65 4225 85 7225
6 36 26 676 46 2116 66 4356 86 7396
7 49 27 729 47 2209 67 4489 87 7569
8 64 28 784 48 2304 68 4624 88 7744
9 81 29 841 49 2401 69 4761 89 7921
10 100 30 900 50 2500 70 4900 90 8100
11 121 31 961 51 2601 71 5041 91 8281
12 144 32 1024 52 2704 72 5184 92 8464
13 169 33 1089 53 2809 73 5329 93 8649
14 196 34 1156 54 2916 74 5476 94 8836
15 225 35 1225 55 3025 75 5625 95 9025
16 256 36 1296 56 3136 76 5776 96 9216
17 289 37 1369 57 3249 77 5929 97 9409
18 324 38 1444 58 3364 78 6084 98 9604
19 361 39 1521 59 3481 79 6241 99 9801
20 400 40 1600 60 3600 80 6400 100 10000

Cubes 1 to 100

The following table shows the cubes from 1 to 100:

Number Cube Number Cube Number Cube Number Cube Number Cube
1 1 21 9261 41 68921 61 226981 81 531441
2 8 22 10648 42 74088 62 238328 82 551368
3 27 23 12167 43 79507 63 250047 83 571787
4 64 24 13824 44 85184 64 262144 84 592704
5 125 25 15625 45 91125 65 274625 85 614125
6 216 26 17576 46 97336 66 287496 86 636056
7 343 27 19683 47 103823 67 300763 87 658503
8 512 28 21952 48 110592 68 314432 88 681472
9 729 29 24389 49 117649 69 328509 89 704969
10 1000 30 27000 50 125000 70 343000 90 729000
11 1331 31 29791 51 132651 71 357911 91 753571
12 1728 32 32768 52 140608 72 373248 92 778688
13 2197 33 35937 53 148877 73 389017 93 804357
14 2744 34 39304 54 157464 74 405224 94 830584
15 3375 35 42875 55 166375 75 421875 95 857375
16 4096 36 46656 56 175616 76 438976 96 884736
17 4913 37 50653 57 185193 77 456533 97 912673
18 5832 38 54872 58 195112 78 474552 98 941192
19 6859 39 59319 59 205379 79 493039 99 970299
20 8000 40 64000 60 216000 80 512000 100 1000000

Also, Check

Solved Examples on Squares and Cubes

Here are some solved examples below:

Example 1: Find the square of number 28.

Solution:

Given number: 28

To find it’s square, multiply it twice:

Square of 28= 28⨯28= 784

The final answer is 784.

Example 2: A square park is being constructed. The length of one side is 45 m. Find the area of the square park

Solution:

Given length: 45 m

Area of park = side2

⇒ Area of park = 452

⇒ Area of park = 45⨯45

⇒ Area of park = 2025 m2

The area of square park is 2025 m2.

Example 3: Find the square root of 144.

Solution:

Given that: 144

Square root of 144 = √144

⇒ Square root of 144 = 12, [because 12⨯12= 144]

Example 4: Determine a cube of 7.

Solution:

Given number: 7

cube of ‘7’ = 73

⇒ cube of ‘7’ = 7⨯7⨯7

⇒ cube of ‘7’ = 343

The cube of ‘7’ is ‘343’.

Example 5: Calculate the volume of a cube when its one edge is given 2 units

Solution:

Given edge: 2 units

Volume of cube = edge3

⇒ Volume of cube = 23

⇒ Volume of cube = 2⨯2⨯2

Volume of cube = 8 unit3

Practice Questions on Squares and Cubes

Here are some practice questions involving squares and cubes:

Question 1. Calculate the squares of the following numbers:

  • (a) 23
  • (b) 44
  • (c) 65

Question 2. Find the cube root of the following integers:

  • (a) 64
  • (b) 512
  • (c) 1331

Question 3. Choose the correct perfect square:

  • (a) 81
  • (b) 125
  • (c) 8

Question 4. Choose the correct perfect cube:

  • (a) 8
  • (b) 9
  • (c) 100

Question 5. A cube whose edge is 8 units. Find the volume of the cube?

Squares and Cubes – FAQs

What is Square Number?

A number which we multiply two times by itself, it becomes a square number.

How can I identify the square numbers?

In some cases, we can identify the squares through the last digit. If it ends with 1,4, 5, 6 or 9, it might be a square number. Also the difference between consecutive square numbers form an A.P.

How can I find a cube of any integers?

Multiply it three times to find the cube number. Using a calculator can also help or you can memorize a cube of numbers to a certain limit.

What is Square Root?

Square root is a reverse process of finding a square number. For example, square root of √169 is 13 because 13⨯13= 169.

What are Cube Numbers?

Cube numbers can be found by multiplying a number thrice by itself.

What will be the Square of Integer 21?

The square of integer 21 will be 441.

Is there any pattern to find square and cube numbers?

Yes, there are many patterns to find any square number and cube number. We can identify these through the difference between consecutive squares and consecutive cubes.


 


Reffered: https://www.geeksforgeeks.org

Class 8

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