When an integer is multiplied by itself, the resultant number is known as its square number. Basically, a square number is a number that is obtained by the product of two same numbers. For example, the square of \(4\) is \(4^{2} = 4 \times 4 = 16\). The square of any number is always positive. If a negative number is multiplied by itself, it results in a positive number. For example, \((-16)^{2} = -16 \times -16 = 256\).

Square \(\textbf{1}\) to \(\textbf{50}\) is the list of squares of all numbers from \(1\) to \(50\). The highest value of squares from \(1\) to \(50\) is \(50^{2}=2500\) and the lowest value is \(1^{2}=1\).

Maths Notes Free PDFs

What is Square 1 to 50?

Square 1to 50 is the list of squares of all numbers from 1to50. The value of squares from 1to50 ranges from 1to 2500.

• Exponent form of square number, (\(x\)): \((x)^{2}\).
• Highest value of squares from \(1\) to \(50\): \((50)^{2} = 2500\).
• Lowest value of squares from \(1\) to \(50\): \((1)^{2} = 1\).

The chart given below shows the squares of numbers from 1 to 50

Note: Memorizing these values will help students to simplify the equations quickly.

The table given below shows the square values of numbers from 1 to 50 for even numbers:

Formula:
(2n)² = 4n²

This means:
Take any number (let's call it n), multiply it by 2 (to get an even number), then square it.

Let’s try with examples:

If n = 1, then:
(2 × 1)² = 2² = 4
Or using the formula: 4 × (1²) = 4 × 1 = 4

If n = 2, then:
(2 × 2)² = 4² = 16
Or using the formula: 4 × (2²) = 4 × 4 = 16

So, by changing the value of n, we can find the squares of all even numbers easily.

Square 1 to 50 of Even Numbers

The table given below shows the square values of numbers from \(1\) to \(50\) for even numbers:

Square 1 to 50 of Odd Numbers

The table given below shows the square values of numbers from 1to50 for odd numbers:

Formula:
(2n + 1)² = 4n(n + 1) + 1

This means:
Take any number (n), plug it into the formula above, and you’ll get the square of an odd number.

Let’s try with examples:

If n = 1, then:
(2 × 1 + 1)² = 3² = 9
Or using the formula: 4 × 1 × (1 + 1) + 1 = 4 × 2 + 1 = 9

If n = 2, then:
(2 × 2 + 1)² = 5² = 25
Or: 4 × 2 × (2 + 1) + 1 = 4 × 6 + 1 = 25
 

So by changing the value of n, you can easily find the square of any odd number.

How to Calculate Values of Squares 1 to 50?

The square1To50 can be found by the following methods which are given below:

• Multiplication by itself
• Using algebraic identities

Method 1: Multiplication by itself

In this method, the number is multiplied by itself, then the product gives us the square of that number. Let us understand this with the help of an example.

For example, the square of \(35 = 35 \times 35 = 1225\). Here, the resultant product “\(1225\)” gives us the square of the number “\(35\)”. This method is good for finding the square of smaller numbers between \(1\) to \(50\).

Method 2: Using algebraic identities

In this method, we use basic algebraic identities to find the square of a number from 1to50.

For example, to find the square of \(34\), we can express \(34^{2}\) as:

• Case 1: \(34^{2}=(30 + 4)^{2} = (30)^{2}+2\times(30)\times(4)+(4^{2})=900+240+16=1156\).
• Case 2: \(34^{2}=(40 – 6)^{2} = (40)^{2}-2\times(40)\times(6)+(6)^{2}=1600-480+36=1156\).

This method is good for finding the square of larger numbers between \(1\) to \(50\).

Square 1 to 50 Table

The table given below shows the square values of numbers from 1to50.

The table given below shows the square values of numbers from \(1\) to \(50\):

Note: These values can be learned and used for mathematical calculations or it is good for students to memorize these squares from 1to50 values thoroughly for faster math calculations.

Properties of Square Numbers

1. Square numbers never end with 2, 3, 7, or 8.
   For example, numbers like 12, 23, 37, or 48 can’t be perfect squares.
    
2. The number of zeros at the end matters.
   If a number ends with an even number of zeros (like 100, 40000), it can be a square. If it ends with an odd number of zeros (like 1000, 500000), it can’t be a perfect square.
    
3. Even and odd numbers stay the same when squared:
   • Even number squared = Even number (e.g., 6² = 36)
      
   • Odd number squared = Odd number (e.g., 7² = 49)
      
4. Square of any natural number (except 1) is either:
    
   • A multiple of 3 (like 9, 36, etc.)
      
   • Or just 1 more than a multiple of 3 (like 4, 10, etc.)
      
5. Same with 4:
    
   • A square number is either a multiple of 4 or 1 more than a multiple of 4.
      
6. The last digit of the square depends on the last digit of the number.
   For example, 4² = 16 and 14² = 196 – both end in 6.
    
7. There are numbers p and q such that:
   p² = 2 × q². This shows a special relationship between some square numbers.
    
8. The difference between the square of (n+1) and n is:
   (n+1)² - n² = (n + 1) + n. For example, (4+1)² - 4² = 25 - 16 = 9 = 4 + 5.
    
9. The square of any number is the sum of the first n odd numbers.
   For example, 4² = 1 + 3 + 5 + 7 = 16.
    
10. For any number greater than 1, (2n, n² – 1, n² + 1) is a Pythagorean triplet.
    Example: If n = 3 → (6, 8, 10) is a Pythagorean triplet.

Square 1 to 50 Solved Examples

Example 1: Two square wooden planks have sides \(20\)m and \(32\)m respectively. Find the combined area of both wooden planks?

Solution: Area of wooden plank = \((side)^{2}\)

\(\Rightarrow\) Area of 1st wooden plank \(= 20^{2} = 400 m^{2}\)

\(\Rightarrow\) Area of 2nd wooden plank \(= 32^{2} = 1024 m^{2}\)

Therefore, the combined area of wooden planks is \(400 + 1024 = 1424 m^{2}\).

Example 2: Find the sum of the first \(50\) odd numbers.

Solution: The sum of first \(n\) odd numbers is given as \(n^{2}\).

\(\Rightarrow\) Sum of first \(50\) odd numbers \((n) = 50^{2}\)

Using values from square \(1\) to \(50\) chart, the sum of the first \(50\) odd numbers \(= 50^{2} = 2500\).

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams. For better practice, solve the below provided previous year papers and mock tests for each of the given entrance exam: