 # How many Rectangles are there on a Chess Board?

To find out the number of rectangles that are on a chessboard, first, we have to find how many squares are there on a chessboard. You can do that either by reading our previous article on How Many Square Are There In A Chessboard or you can just read it here and then further find out the number of rectangles (step-by-step) that are there on a chessboard.

Squares on a chess board

There are more squares in a chessboard than 64 (1 × 1) squares.

The square count starts from (1 x 1) all the way up to (8 × 8).

1. The number of squares after multiplying (1 × 1) squares with 8 squares across the length and 8 squares along the width = (8 × 8) = 64 squares.
2. The number of squares after multiplying (2 × 2) squares with the size of the square increasing by 1 square as the number of squares across the length will be down to 7 squares and the ones along the width will also decrease to 7 squares. So, there are (7 × 7 )= 49 (2 x 2) squares.
3. The number of squares after multiplying (3 × 3) squares with 6 squares across the length and 6 along the width, the number of squares are (6 × 6) = 36 (3 x3) squares.
4. The number of squares after multiplying (4 × 4) squares with 5 squares across the length and 5 along the width = (5 × 5) = 25 (4 × 4) squares.
5. The number of squares after multiplying (5 × 5) squares with 4 squares across the length and 4 along the width is = (4 × 4) = 16 (5 × 5) squares.
6. The number of squares after multiplying (6 × 6 squares) with 3 squares across the length and 3 along the width = (3 × 3)= 9 (6 × 6) squares.
7. The number of squares after multiplying (7 × 7) squares with 2 squares across the length and 2 along the width = (2 × 2) = 4 (7 × 7) squares.
8. The number of squares after multiplying (8 × 8) squares with 1 square across the length and 1 along the width = (1 × 1) = 1 (8 × 8) squares.

Therefore, the total number of squares on the chessboard will be by adding all the numbers above:-

[ 64 + 49 + 36 + 35 + 16 + 9 + 4 + 1 ]

By adding the numbers we get, 204 which is the number of squares on a chessboard.

Now, let’s find out the total number of rectangles that are present in the (8*8) chessboard. ## How many Rectangles are there on a chess board

How Many Rectangles Are There on A Chessboard” is more of a math question than a chess question.

Now, let’s take a look at how to calculate the number of rectangles in an 8″ x 8″ chessboard.

There are 9 vertical lines and 9 horizontal lines on the chess board.

To form a rectangle you must choose 2 of the 9 vertical lines and 2 of the 9 horizontal lines. So the number of rectangles is (9C2 x 9C2) which is equal to 1296 rectangles.

Let’s tell you the number of rectangles in a chessboard step-by-step:-

• The number of ways to select Rows(R) x Columns (C) for a rectangle on a chessboard is (9-R) x (9-C). Rectangles with one row and many columns.
• (1 x 1) rectangles = (9–1)*(9–1) = 64
• (1 x 2) rectangles = (9–1)*(9–2) = 56
• (1 x 3) rectangles = (9–1)*(9–3) = 48
• (1 x 4) rectangles = 5 x 8 = 40
• (1 x 5) rectangles = 4 x 8 = 32
• (1 x 6) rectangles = 3 x8 = 24
• (1 x 7) rectangles = 2 x 8 = 16
• (1 x 8) rectangles = 1 x 8 = 08

Now, let’s add up the numbers.

( 64+56+48+40+32+24+16+8 = 288)

If you see there is a pattern here,

Total number of one row rectangles = (8*1)+ (8*2) +…+(8*8)  = 8 * (1+2+3+4+5+6+7+8)

= 8 * (8*9/2) Since,∑n = n(n+1)/2

=8 * 36

Similarly, Two row rectangles= (7*1)+(7*2)+….+(7*8)

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

=7* (7*8/2) Since,∑n = n(n+1)/2

=7 * 36

Three row rectangles= (6*1)+(6*2)+(6*3)+….+(6*8)

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

= 6* (6*7/2) Since, ∑n = n(n+1)/2

= 6*36

Four row rectangles= (5*1)+(5*2)+…+(5*8)

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

= 5* (5*6/2) Since, ∑n = n(n+1)/2

= 5*36

Five row rectangles = (4*1)+(4*2)+…+(4*8)

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

=4* (4*5/2) Since, ∑n = n(n+1)/2

= 4*36

Six row rectangle = (3*1)+(3*2)+…(3*8)

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

= 3* (3*4/2) Since, ∑n = n(n+1)/2

=3*36

Seven row rectangle = (2*1)+(2*2)+….+(2*8)

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

= 2* (2*3/2) Since, ∑n = n(n+1)/2

= 2*36

Eight row rectangle = (1*1)+(1*2)+….+(1*8)

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

= 1* (1*2/2) Since, ∑n = n(n+1)/2

= 1*36

So total number of rows rectangles = (8*36) + (7*36) + (6*36) + (5*36) + (4*36) + (3*36) + (2*36) + (1*36)

= 36 * (1+2+3+4+5+6+7+8)

= 36 * 36

= 1296

Therefore the total number of rectangles is 1296.

But every square is also a rectangle.

Therefore,

8×8 square = 1 Since, { (9–8) * (9–8) }

7×7 squares = 4

6×6 squares = 9

5×5 squares = 16

4×4 squares = 25

3×3 squares = 36

2×2 squares = 49

1×1 squares = 64

Total squares only = 1+4+9+16+25+36+49+64 = 204

Total square and rectangles = 1296

Total rectangles only = 1296 – 204 = 1092

Thus, the total number of rectangles only is 1092.

Takeaway

You found out the total number of squares in a chessboard, the total number of squares and rectangles in a chessboard, and lastly the total number of rectangles only in a chessboard which is 1092. You must have also learned that every square is also a rectangle.

Hope we were able to give you answers to the questions that you were looking for.