Question

If the circumference of a circle and the perimeter of a square are equal, then ______.

Options

• Area of the circle = Area of the square

• Area of the circle > Area of the square

• Area of the circle < Area of the square

• Nothing definite can be said about the relation between the areas of the circle and square.

• None of these

Answer 1

If the circumference of a circle and the perimeter of a square are equal, then Area of the circle > Area of the square.

Explanation:

According to the given condition,

Circumference of a circle = Perimeter of square

2πr = 4a  ...[Where r and a are radius of circle and side of square respectively]

⇒ `22/7 "r"` = 2a

⇒ 11r = 7a

⇒ a = `11/7 "r"`

⇒ r = `(7"a")/11`  ...(i)

Now, Area of circle,

A₁ = πr²

= `π((7"a")/11)^2`   ...[From equation (i)]

= `22/7 xx (49"a"^2)/121` 

= `(14"a"^2)/11`  ...(ii)

And Area of square,

A₂ = (a)²  ...(iii)

From equations (ii) and (iii),

A₁ = `14/11 "A"_2`

∴  A₁ > A₂

Hence, Area of the circle > Area of the square.