A wire is in the shape of a rectangle. Its length is $40\ cm$ and breadth is $22\ cm$. If the same wire is rebent in the shape of a square, what will be the measure of each side. Also find which shape encloses more area?

Given: A wire is in the shape of a rectangle. Its length is $40\ cm$ and breadth is $22\ cm$.

To do: To find the measure of each side when the same wire is rebent in the shape of a square and also find which shape encloses more area.

Solution: As given here, length $l=40\ cm$

Breadth $b=22\ cm$

Area of the rectangle $A_{rectangle}=l\times b$

$=40\ cm\times 22\ cm$

$=880\ cm^2$

We know the formula for the perimeter of a rectangle $P_{rectangle}=2(l+b)$

Or $P_{rectangle}=2(40\ cm+22\ cm)$

Or $P_{rectangle}=2(62\ cm)$

Or $P_{rectangle}=124\ cm$

Let "$a$" by the side of the square formed on rebending the wire. The perimeter will be the same.

Perimeter of the square $P_{square}=4a$

So, $4a=124\ cm$

Or $a=\frac{124}{4}$

Or $a=31\ cm$

Area of the square $A_{square}=a^2$

$=31^2$

$=961\ cm^2$

Here, on comparing the areas of both shapes, we find

$A_{square}$>$A_{rectangle}$

So, the shape of the square encloses more area than the shape of the rectangle.

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