Question

A wire is in the shape of a square of side 10 cm. If the wire is re-bent into a rectangle of length 12 cm, find its breadth. Which encloses more area- the square or the rectangle?

Answer 1

Hint: We will first find the perimeter of the square using the formula, $4s$, where $s$ is the side of the square. The perimeter of the rectangle will also be equal. Therefore, we can calculate the breadth of the rectangle using the formula of the perimeter of the rectangle, $2\left( {l + b} \right)$, where \[l\] is the length of the rectangle and $b$ is the breadth of the rectangle. Also, find the area of the square and rectangle and see whose area is greater.

Complete step-by-step answer:
We are given that a wire is in the shape of a square of side 10 cm.

We can calculate the total length of the wire by finding the perimeter of the square.
We know that the perimeter of the square is $4s$, where $s$ is the side of the square.
Then, the length of the wire is $4\left( {10} \right) = 40cm$
Now, this will also be equal to the perimeter of the rectangle formed of length 12cm.

It is known that the perimeter of the rectangle is $2\left( {l + b} \right)$, where \[l\] is the length of the rectangle and $b$ is the breadth of the rectangle.
Hence,
$
  40 = 2\left( {12 + b} \right) \\
   \Rightarrow 40 = 24 + 2b \\
   \Rightarrow 40 - 24 = 2b \\
   \Rightarrow 16 = 2b \\
$
Divide the equation by 2.
$b = 8$
Hence, the breadth of the triangle is 8m.
Now, we will find the area of the formed square and rectangle.
Area of the square is given as ${s^2} = {\left( {10} \right)^2} = 100c{m^2}$
Whereas the area of the rectangle is $l \times b = 12 \times 8 = 96c{m^2}$
Hence, the area of the square is more than the area of the rectangle.

Note: Perimeter of the figure is the length of the boundary of the figure. Thus, if the wire of the same length is used to make both the shapes, their perimeter will be equal. Students must know the formulas of perimeter and area of rectangle and square to do these type of questions.