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
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.
= 15^2 = 125.
Number of squares and rectangles in a grid
Reviewed by goodinfo
on
October 13, 2019
Rating:
Very useful tricks
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