Question

The areas of a square and a rectangle are equal. will the perimeter of the square be more or the perimeter of the rectangle?

Answer 1

Answer :
Given : The areas of a square and a rectangle are equal.
And we know Area of square = ( Side length of square )²
And
Area of rectangle = Length $\times$ Breadth
So, from given condition , we get

( Side length of square )² = Length $\times$ Breadth

Side length of square = $\sqrt{Length\times Breadth}$
And we know Perimeter of square = 4 ( Side length of square ) , So

Perimeter of square = 4 ( $\sqrt{Length\times Breadth}$ )
Taking whole square on both hand side , we get
( Perimeter of square )² = 16( Length $\times$ Breadth )
And
we know Perimeter of rectangle = 2 ( Length + Breadth )
Taking whole square on both hand side , we get
( Perimeter of rectangle )² = 4( Length² + Breadth² + 2 Length $\times$ Breadth )

Now we can check for different values of length and breadth of rectangle , As :

Let Length = 5 cm and Breadth = 2 cm , So

( Perimeter of square )² = 16( 5 $\times$ 2) = 160 cm
And
( Perimeter of rectangle )² = 4( 5² + 2² + 2 $\times$ 5 $\times$ 2 ) = 4 ( 25 + 4 + 20 ) = 196 cm

Or

Let Length = 10 cm and Breadth = 6 cm , So

( Perimeter of square )² = 16( 10 $\times$ 6) = 960 cm
And
( Perimeter of rectangle )² = 4( 10² + 6² + 2 $\times$ 10 $\times$ 6 ) = 4 ( 100 + 36+ 120 ) = 1024 cm

So, we can say that

When areas of a square and a rectangle are equal. Then perimeter of the rectangle be more than the perimeter of the square . ( Ans )