Question

There is only one square whose perimeter and area are the same. What is the length of each side of this square?
The other rectangles whose perimeter and area are the same. Only one of these rectangles has a length and width which are whole numbers. What are the length and width of this rectangle ?

Answer 1

Let side of the square be $l$.

then from the question

$4l=l^2$

$l=4$

Let length and width of the rectangle be $a$ and $b$

then

from the question

$2(a+b)=ab$

$ab-2a=ab$

$a=\frac{2b}{b-2}$

keep $b=3$ and $b=4$ then we get $a=6$and $a=4$

Since, $a=b=4$ is a square

we take a perfect rectangle $a=6$ and $b=3$