Squares and Cubes

Squares and Cubes are such numbers which we multiply twice and thrice with themselves respectively. In this article, we will explore the properties and patterns of solving squares and cubes. Comprehending what squares and cubes are and studying the functions of squares and cubes through tables and charts is the primary objective of this article. To make our concepts strong, we will practice different types of questions. They are very important for foundational understanding.

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.

What is Square Number?

When an integer is multiplied by itself, it is called a square 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 Number?

When we multiply an integer three times by itself, it is called a cube 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’s 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 number of ‘3’.

Also, we can simply multiply it thrice to find its cube number. 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

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 which 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 which can be multiplied by itself thrice. 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

Solved Examples on Squares and Cubes

After learning the basics of squares and cubes, we will see some solved examples to make our topic stronger. Here are some solved examples below:

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.

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.

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]

4. Determine 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’.

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 Problems on Squares and Cubes

To get perfect in any topic, practice questions are necessary. Here are some practice questions involving square and cubes:

1. Calculate the squares of the following numbers:

(a) 23 (b) 44 (c) 65

2. Find the cube root of following integers:

(a) 64 (b) 512 (c) 1331

3. Choose the correct perfect square:

(a) 81 (b) 125 (c) 8 (d) 27

4. Choose the correct perfect cube:

(a) 8 (b) 9 (c) 25 (d) 100

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

Squares and Cubes – FAQs

These are some frequently asked questions related to squares and cubes.

1. What is Square Number?

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

2. 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.

3. 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.

4. 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.

5. What are Cube Numbers?

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

6. What will be the Square of Integer 21?

The square of integer 21 will be 441.

7. 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.