In this article discussed about formulas to find **number of squares** and rectangles in given figure of of ‘**n’** number of rows and **‘m’** is the number of columns

## Count number of squares and number of rectangles in a given N x M Grid | How to count square in reasoning

Contents

### How many squares are there in the given figure

**How many squares are there in the figure of same number of rows and columns**

( i.e Number of squares in square grid )

**Example – 1** – How many squares are there in an 4 x 4 grid

**Solution** : There are 4 rows and 4 columns in the above figure. So let n =4

Here we using two types of formulas for finding number of squares in an n x n grid as follows

**Formula – 1**

n^{2} + (n -1 )^{2} + (n-2)^{2} + – – – – – + (n – n)^{2 }

Now substitute n = 4 in the above formula

= 4^{2} + ( 4-1)^{2} + (4- 2 )^{2 }+ (4 – 3)^{2 }+ (4 – 4)^{2 }

= 16 + 9 + 4 + 1 + 0 = 30

**Formula – 2**

Apply the formula

Substitute n = 4 in above formula

= 4 x 5 x 9 / 6 = 30

So number of squares in an 4 x 4 grid is 30

**Example -2 **How many squares are there in an 5 x 5 grid

**Solution** : There are 5 rows and 5 columns in the above figure. Hence n=5

**Formula
**

n^{2} + (n -1 )^{2} + (n-2)^{2} + – – – – – + (n – n)^{2 }

So according to above substitute n = 5

= 5^{2} + ( 5-1)^{2} + (5- 2 )^{2 }+ (5 – 3)^{2 }+ (5 – 4)^{2 }+ (5 – 5)^{2 }

= 25 + 16 + 9 + 4 + 1 + 0 = 55

Number of squares in an 5 x 5 grid is 55

**How many squares are there in the figure of ‘n’ number of rows and ‘m’ number of columns**

( i.e Number of squares in rectangle grid )

**Example – 3 **How many squares are there in an 3 x 4 grid

**Solution** : There are 4 rows and 5 columns in the above figure.

Let number of rows ( n)= 4 & number of columns (m) = 5

Here we using simple formulas as follows

**Formula- 1**

( n x m ) + (n -1 ) (m – 1) + (n-2 ) ( m- 2) + – – – – – + (n – n ) or (m – m)^{
}

Now substitute n = 5 and m = 4

= ( 4 x 5) + ( 4 – 1) (5 – 1 ) + ( 4 – 2) (5 – 2 ) + ( 4 – 3) (5 – 3 ) + ( 4 – 4) (5 – 4 )

= 20 + 12 + 6 + 2 +0 = 40

**Formula- 2**

Here consider large value is n

i.e n = 5 and m = 4, Now substitute these values in above formulas

= [ 4 x 1 x 5 / 2 ] + [ 4 x 5 x 9 / 6 ]

= 10 + 30 = 40

Number of squares in given figure = 40

### How many rectangles are there in the given figure

**Count number of rectangles in the figure of same number of rows and columns grid
**

( i.e __Counting rectangles within a square )__

**Example – 4 **How many rectangles are there in an 5 x 5 grid

Solution: There are 5 rows and 5 columns in the above figure.

Let number of rows or columns ( n)=5

Here for finding the rectangles there are having two methods

**Formula – 1**

[n + ( n -1 ) + ( n -2 ) + ( n -3 ) + – – – – – + ( n -n )]^{2}

Now substitute the values of’ ** n**‘ in the above formula

= [ 5 + (5 – 1 ) + (5 – 2 ) + (5 – 3 ) + (5 – 4 ) + (5 – 5) ]^{2}

= [ 5 + 4 + 3+ 2 + 1 +0 ]^{2}

= 15 x 15 = 225

**Formula – 2**

To count the number of rectangles within a square by using formula of

Now substitute the values of’ ** n**‘ in the above formula

= [ 5 ( 5 + 1 ) /2 ] ^{2 }=[ 5 x 6 / 2 ]^{2}

= 15 ^{2} = 225

Total number of rectangles in given figure =** 225**

**Count number of rectangles in the figure of **** ‘n’ number of rows and ‘m’ number of columns**

( i.e __Counting rectangles within a rectangle ) __

**Example – 5 **How many rectangles are there in an 4 x 5 grid

**Solution: **There are 4 rows and 5 columns in the above figure.

Let number of rows ( n)=4 & number of columns (m) = 5

Here for finding the rectangles there having two methods

**Formula – 1** ( Formula for finding number of rectangles in figure of ‘n’ number of rows and ‘m’ is the number of columns)

[n + ( n -1 ) + ( n -2 ) + ( n -3 ) + – – – – – + ( n -n )] x [m + ( m -1 ) + (m -2 ) + (m-3 ) + – – – – – + ( m -m )]

Now substitute the values of’ ** n**‘ and ‘** m**‘ in the above formula

= [ 4 + (4 – 1 ) + (4 – 2 ) + (4 – 3 ) + (4 – 4 )] x [ 5 + (5 – 1 ) + (5 – 2 ) + (5 – 3 ) + (5 – 4 ) + (5 – 5) ]

= [ 4 +3 +2 +1 +0 ] x [ 5 + 4 + 3+ 2 + 1 +0 ]

= 10 x 15 = 150

**Formula – 2 **

Now substitute the values of’ ** n**‘ and ‘** m**‘ in the above formula

= 4 x 5 x 5 x 6 / 4 = 150

Total number of rectangles in given figure = 150

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