Squares and Cubes (original) (raw)

Last Updated : 23 Jul, 2025

**Squares and Cubes are mathematical operations involving numbers that are essential in various areas of mathematics. The square of a number is obtained by multiplying the number by itself (i.e., n2 = n × n), while the cube of a number is obtained by multiplying the number by itself twice (i.e., n3 = n × n × n)

A square is used to find the area of a 2d figure, whereas a cube is used to find the volume of a 3d figure.

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Square and Cube

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.

Table of Content

Square of a Number
Cube of a Number
Perfect Square and Cubes
Squares and Cubes from 1 to 50
Chart of Squares and Cubes 1 to 100

Square of a Number

When an integer is multiplied by itself, it is called the square of that number. In simple words, a number that is multiplied by two 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

Cube of a Number

When we multiply an integer by itself three times, 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 examples are as follows:

23 = 8

113 = 1331

Square and Cube 1 to 20

In this, we will learn the squares and cubes of numbers from 1 to 20. Let's have a look at them

Number	Square(n2)	Cube (n3)	Number	Square (n2)	Cube (n3)
1	1	1	11	121	1331
2	4	8	12	144	1728
3	9	27	13	169	2197
4	16	64	14	196	2744
5	25	125	15	225	3375
6	36	216	16	256	4096
7	49	343	17	289	4913
8	64	512	18	324	5832
9	81	729	19	361	6859
10	100	1000	20	400	8000

Chart of Squares and Cubes

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 squares 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 that help them calculate easily and quickly. Here is the table in which the squares and cubes from 1 to 30 are given:

**Read: **Squares 1 to 30

Square-1-to-30

**Read: **Cubes 1 to 30

Cubes-1-to-30

Perfect Squares 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 twice. 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 cubes, we learn some easy ways for their identification. First, we learn about the '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 a perfect square and a 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 thismethod, 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

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

**Patterns in Square Numbers

Square-Number

Patterns of Square Numbers

There are various patterns in the square numbers, some of which 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 the 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

cube_number_pattern

Cube number pattern

Some common 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 Question on Squares and Cubes

Here are some solved examples below:

**Question 1: Find the square of the 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 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:

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

64
512
1331

**Question 3. Choose the correct perfect square:

**Question 4. Choose the correct perfect cube:

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

**Question 6. Find the square of the number 17.

**Question 7. Determine the cube of 5.

**Question 8. What is the square root of 121?

**Question 9. Find the perfect cube among the following numbers:

**Question 10. Calculate the volume of a cube with a side length of 10 units.

**Conclusion

Understanding squares and cubes is essential for various mathematical and practical applications. The Squares involve multiplying a number by itself, while cubes involve raising a number to the power of three. Mastery of these concepts is crucial for solving problems in algebra, geometry, and real-world scenarios. The Practice with these questions helps reinforce the fundamental skills needed to perform the calculations accurately and efficiently.