Question

If a wire in the shape of a square is re-bent into a rectangle, then the ____ of both shapes remain same, but _____ may vary.

• A. perimeter
• B. area

Answer 1

Consider the perimeter and area of a square and rectangle and compare its values

Since the length of the wire remains same throughout, the length of the boundary of both the square and rectangle will remain the same.

Hence the perimeter of the of both shapes shall remain the same but the area will change.

For example:

Step 1: Use formula for perimeter of square and rectangle

Consider a wire of length $100cm$ is in the shape of the square

$Perimeterofsquare=4\times \left(lengthofside\right)$

Hence each side of the square is of $25cm$

$Perimeterofrec\tan gle=2\left(length+width\right)$

As perimeter remains constant,

$length+width=50cm$

Now suppose this wire is bent into a rectangle with length $30cm$ and width $20cm$

Step 2: Use formula for area of square and rectangle

$Areaofsquare={\left(lengthofside\right)}^2$

$={25}^2=625c{m}^2$

Thus the area of the square region is $625c{m}^2$

$Areaofsquare=length\times width$

$=30\times 20=600c{m}^2$

Hence the area of the region now becomes $600c{m}^2$

From the above example we can conclude that the area may vary but the perimeter remains the same when a square is bent into a rectangle.