Question

A wire is in the shape of a square of side 10 cm. If the wire is rebent into a rectangle of
length 12 cm, find its breadth. Which encloses more area, the square or the rectangle
and by how much?​

Answer 1

Breadth is 8 cm

Square encloses more area thn rectangle by 4cm²

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Answer 2

To Find :

• Breadth of the Rectangle .

• The shape with greater Area.

Given :

• Side of the Square = 10 cm

• Length of the Rectangle = 12 cm

We Know :

$\bullet$ Perimeter of a Square :

$\bigstar \purple{\bf{P = 4 \times Side}}$

$\bullet$ Area of a Square :

$\bigstar \purple{\bf{A = (Side)^{2}}}$

$\bullet$ Perimeter of a Rectangle :

$\bigstar \purple{\bf{P = 2(length + Breadth)}}$

$\bullet$ Area of a Rectangle :

$\bigstar \purple{\bf{A = length \times Breadth}}$

AnalYsis :

• Since both the shapes (Square and Rectangle) are made from the same wire , the perimeter will be same in both the cases.. i.e,

$\therefore \sf{Perimeter_{(Square)} = Perimeter_{(Rectangle)}}$

Hence by calculating the perimeter of the square , we can find the breadth , by using the given Length.

• As it is asked that which shape incloses more area and by how much , we can find the individual area and then by finding the difference between them , we can determine that which shape exceeds the area of the other shape.

Solution :

"Perimeter of the SQuare"

Given,

• side = 10 cm

Using the formula for Perimeter of a Square , and by substituting the value in it , we get :

$\implies \sf{P = 4 \times Side} \\ \\ \implies \sf{P = 4 \times 10} \\ \\ \implies \sf{P = 40 cm} \\ \\ \therefore \blue{\sf{Perimeter = 40 cm}}$

Hence , the Perimeter of the Square is 40 cm.

"Breadth of the RectanGle"

Given :

• Length = 12 cm

• Perimeter = 40 cm

$\boxed{\sf{Perimeter_{(Square)} = Perimeter_{(Rectangle)}}}$

Taken :

Let the Breadth be x cm.

Using the formula for Perimeter of a Rectangle and by substituting the values in it , we get :

$\implies \sf{P = 2(length + Breadth)} \\ \\ \implies \sf{40 = 2(12 + x)} \\ \\ \implies \sf{40 = 24 + 2x} \\ \\ \implies \sf{40 - 24 = 2x} \\ \\ \implies \sf{16 = 2x} \\ \\ \implies \sf{\dfrac{\cancel{16}}{\cancel{2}} = x} \\ \\ \implies \sf{8 cm = x} \\ \\ \therefore \purple{\sf{x = 8 cm}}$

Hence , the Breadth of the Rectangle is 8 cm.

"Area of the SQuare"

Given :

• Side = 10 cm

Using the formula for Area of a Square and by substituting the values in it ,we get :

$\implies \sf{A = (Side)^{2}} \\ \\ \implies \sf{A = (10)^{2}} \\ \\ \implies \sf{A = 100 cm^{2}} \\ \\ \therefore \purple{\sf{A = 100 cm^{2}}}$

Hence , the Area of the Square is 100 cm².

"Area of the RectanGle"

Given :

• Length = 12 cm

• Breadth = 8 cm

Using the formula for Area of a Rectangle , and by substituting the values in it , we get :

$\implies \sf{A = length \times Breadth} \\ \\ \implies \sf{A = 12 \times 8} \\ \\ \implies \sf{A = 96 cm^{2}} \\ \\ \therefore \purple{\sf{A = 96 cm^{2}}}$

Hence , the Area of the Rectangle is 96 cm ².

Hence , The Area of the Square is Greater than the Area of the Rectangle .

Difference between the Area of the Square and Area of Rectangle :

$\implies \sf{100 - 96} \\ \\ \implies \sf{4 cm^{2}} \\ \\ \therefore \sf{4 cm^{2}}$

Hence , the Area of Square encloses more the Area of Rectangle by 4 cm².