Question

How many rectangles does a $3\times 3$ square has?

• A. $6^2$
• B. ${10}^2$
• C. $7^2$
• D. $8^2$

Answer 1

The correct option is A $6^2$


Along the top row of the $3\times 3$ square,
Total number of rectangle of size $1\times 1=3$
Total number of rectangle of size $1\times 2=2$
Total number of rectangle of size $1\times 3=1$
Total Number of rectangles in first row = 6
Now, along the first column,
Total number of rectangle of size $1\times 1=3$
Total number of rectangle of size $1\times 2=2$
Total number of rectangle of size $1\times 3=1$
Total Number of rectangles in first column = 6
Hence the total number of rectangle in $3\times 3$ square grid $=6\times 6=6^2$
Now $6^2=1^3+2^3+3^3$
Therefore, total no of rectangles in $3\times 3$ square grid $=1^3+2^3+3^3$

Alternative Solution:

We know that $n\times n$ square has $1^3+2^3+3^3+.....+n^3$ rectangles
Similarly, $3\times 3$ square grid has $1^3+2^3+3^3$ rectangles.