- Total number of the squares in a square. For example,
**find number of square in a chess board** - Total number of the squares in a rectangle
- Total number of rectangles in a square
- Total number of rectangles in a rectangle

## How to find total number of squares in a rectangle

**Formula to find number of squares in a square of size 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:

No. of squares of 1*1 size = 5*4 = 20

No. of squares of 2*2 size = 4*3 = 12

No. of squares of 3*3 size = 3*2 = 6

No. of squares of 4*4 size = 2*1 = 2

So, Total Number of squares of 5*5 size =

= 20 + 12 + 6 + 2 = 40

**Also know: How to find the unit digit when a number is raised to a large power using cyclicity of numbers**

## How to find total number of 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, you already know that Total number of squares in a rectangle of size 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 number of squares in a square of size 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^{2} + 7^{2} + 6^{2} + 5^{2} + 4^{2} + 3^{2} + 2^{2} + 1^{2} =

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

In more detail:

No. of squares of size 1*1 = 8*8 = 64

No. of squares of size 2*2 = 7*7 = 49

No. of squares of size 3*3 = 6*6 = 36

No. of squares of size 4*4 = 5*5 = 25

No. of squares of size 5*5 = 4*4 = 16

No. of squares of size 6*6 = 3*3 = 9

No. of squares of size 7*7 = 2*2 = 4

No. of squares of size 8*8 = 1*1 = 1

So, Total number of squares in a chess board =

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

### Total number of rectangles in a rectangle:

Formula Number of rectangles in a rectangle of size m*n = ^{m}c_{2} * ^{n}c_{2 }Here, C represents combination.

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

**Number of rectangles in a rectangle of size 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

### Total number of rectangles in a square:

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 size 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}^{ }= 225.

#### Summary

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