Number of squares and rectangles in a grid

Find the total number of squares or rectangle in a square or rectangle.

After reading this you will be able to solve following type of problems:

• Total number of the squares in a square. For example, find no. of square in a chess board

• Total number of the squares in a square
• Total number of rectangles in a rectangle
• Total number of squares in a rectangle

Total number of the squares in a rectangle of side m*n =

 m*n + (m-1)*(n-1) + (m-2)* (n-2) + …………( stop when m or n become zero)

Example: Find the total no. of squares in a rectangle of 5*4

Total no of the squares in the 5*4 rectangle = (5*4) + (4*3) + (3*2) + (2*1)

= 20 + 12 + 6 + 2 = 40

Explanation:

Total no of square of 1*1 = 5*4 = 20

Total no of square of 2*2 = 4*3 = 12

Total no of square of 3*3 = 3*2 = 6

Total no of square of 4*4 = 2*1 = 2

So Total no of square = 20 + 12 + 6 + 2 = 40

Total no of the squares in a square

We know that a square is nothing but a rectangle having length equal to breadth.

So, the method will be same here as of rectangle.

Now, We already know that Total no. of square in a rectangle of side m*n =

m*n + (m-1)*(n-1) + (m-2)* (n-2) + …………( stop when m or n become zero)

but for square m = n

putting ‘n’ in place of ‘m’ in above formula, the formula become 

= n*n + (n-1)*(n-1) + (n-2)* (n-2) + ……………….

So, Total no of square in a square of n*n = n^2 + (n-1)^2 + (n-2)^2 +..................

Example: find the total no. of the squares in a Chess board

We all know that a chess board has dimension of 8*8, so we must find the total no. of squares in a square of 8*8.

Total no. of square in a square of 8*8 = 

(8*8) + (7*7) + (6*6) + (5*5) + (4*4) + (3*3) + ( 2*2) + (1*1) =

= 64+ 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204

In more detail:

Total no of square of 1*1 = 8*8 = 64

Total no of square of 1*1 = 7*7 = 49

Total no of square of 1*1 = 6*6 = 36

Total no of square of 1*1 = 5*5 = 25

Total no of square of 2*2 = 4*4 = 16

Total no of square of 3*3 = 3*3 = 9

Total no of square of 4*4 = 2*2 = 4

Total no of square of 5*5 = 1*1 = 1

So, Total no of square in a chess board  =  64+ 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204

Number of rectangles in a rectangle of m*n

Formula Number of rectangles in a rectangle of m*n = ^mc₂ * ⁿc₂                             

Here C represents combination.

If you don’t know combination don’t worry remember the formula written below

Number of rectangles in a rectangle of m*n = [m*(m+1)/2] * [n*(n+1)/2] 

Example: find the total no. of the rectangle in a rectangle of 5*6

Solution: Total no. of rectangle in a rectangle of 5*6 = [5* (5+1)/2] * [6*(6+1)/2]

= [5*6/2] * [ 6*7/2]

= 15 * 21 = 315

Number of rectangles in a square

We know that a square is nothing but a rectangle having length equal to breadth.

So, the method will be same here as of rectangle

We already know that Total no. of rectangle in a rectangle of side m*n =

= [m*(m+1)/2] * [n*(n+1)/2]                             

but for square, m = n

putting ‘n’ in place of ‘m’ in above formula, the formula become

= [n*(n+1)/2] * [n*(n+1)/2] =  [n*(n+1)/2] ^2  

So, Number of rectangles in a square of side length ‘n’ = [n*(n+1)/2] ^2  

 

Example: find the total no. of rectangles in a square having side length equal to 5.

Solution: [5*(5+1)/2] ^2 = [5*6/2]^2
= 15^2 = 125.

Number of square in grid