Question

if the perimeter of square and rectangle are equal, which of the two has greater area

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Answer 1

Answer:

area of square will be greater

Step-by-step explanation:

Let side of square  = s , Length of rectangle  = l , Breadth  = b  ,

We know perimeter of square  = 4 ( Side )  

Perimeter of rectangle = 2 ( Length + Breadth ) ,

So from given condition we get

4 ( a ) = 2 ( l + b  )

2 a = ( l + b  )

a = (l + b)2l + b2                                         --- ( 1 )

We know : Area of square  = ( Side )2  

Area of rectangle = Length ×× Breadth ,

So

,

Area of square = a2

= ((l + b)2)2

= (l + b)24

= l2 + b2 + 2 l b4l + b22

= l + b24

= l2 + b2 + 2 l b4                 ( By substituting value from equation 1 )

And

Area of rectangle  = l b

Now to check which one is grater we subtract area of rectangle from area of square and get :

Area of square - Area of rectangle :

l2 + b2 + 2 l b4 − l b

⇒l2 + b2 + 2 l b − 4 l b 4

⇒l2 + b2 − 2 l b 4

⇒( l − b )24l2 + b2 + 2 l b4 - l b

⇒l2 + b2 + 2 l b - 4 l b 4

⇒l2 + b2 - 2 l b 4

⇒ l - b 24

From above equation we can see that value of ( l - b )2 is remain positive

( As that term have whole square in it )

So,

The whole term (l − b )24l - b 24is positive .

So

Area of square - Area of rectangle = Positive value

Therefore,

Area of square >  Area of rectangle

hope it helps,

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