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    You are at:Home»Math Reasoning»How many squares and rectangles are there in this picture | All Math Tricks
    Count number of squares and number of retangles in a given n x m Grid | Count number of rectangle in a rectangle | How many squares are there in an N x M grid

    How many squares and rectangles are there in this picture | All Math Tricks

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    By sivaalluri on April 7, 2019 Math Reasoning

    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

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    •  Count number of squares and number of rectangles in a given N x M Grid | How to count square in reasoning
      • How many squares are there in the given figure
      • How many rectangles are there in the given figure

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

    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

    n2 + (n -1 )2 + (n-2)2 + – – – – – + (n – n)2

    Now substitute n = 4 in the above formula

    = 42 + ( 4-1)2  + (4- 2 )2 + (4 – 3)2   + (4 – 4)2

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

    Formula – 2

    Apply the formula    \frac{n (n+1) (2n + 1)}{6}

    Substitute n = 4 in the above formula

    = 4 x 5 x 9 / 6 = 30

    So the number of squares in a 4 x 4 grid is 30

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

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

    Formula

    n2 + (n -1 )2 + (n-2)2 + – – – – – + (n – n)2

    So according to the above substitute n = 5

    = 52 + ( 5-1)2 + (5- 2 )2 + (5 – 3)2 + (5 – 4)2+ (5 – 5)2

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

    The number of squares in a 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 a rectangle grid )

    Example – 3 How many squares are there in a 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 use 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

     \left[ \frac{m \, (n - m) \, (m+1)}{2} \right] + \left[ \frac{m \, (m+1) \, (2m+1)}{6} \right]

    Here consider large value is n

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

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

    = 10 + 30 = 40

    The number of squares in given figure = 40

    How many rectangles are there in the given figure

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

    ( i.e Counting rectangles within a square )

    Example – 4   How many rectangles are there in a 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 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 a formula of  \left[ \frac{n \, (n+1)}{2} \right]^2

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

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

    = 15 2 = 225

    The total number of rectangles in the 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 a 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 are two methods

    Formula – 1 ( Formula for finding a number of rectangles in the 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  

     \left[ \frac{n \, m \, (n+1) \, (m+1)}{4} \right]

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

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

    The total number of rectangles in the given figure =  150

     

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