Question

If the circumference of a circle and the perimeter of a square are equal, then which of the following statement is correct?

• A. Area of the circle $=$ Area of the square
• B. Area of the circle > Area of the square
• C. Area of the circle < Area of the square
• D. Nothing definite can be said about the relation between the areas of the circle and square.

Answer 1

The correct option is B Area of the circle > Area of the square

Let the radius of the circle be$r.$

$\therefore$ Area of Circle= $\pi r^2$.

and the circumference=$2\pi r$

The perimeter of the square is equal to circumference of circle.

$\therefore$ Its one side $=\frac{1}{4}\times 2\pi r=\frac{\pi r}{2}$.

So Area of Square $={\left(\frac{\pi r}{2}\right)}^2.$

$\therefore \frac{AreaOfcircle}{AreaOfSquare}=\pi r^2:{\left(\frac{\pi r}{2}\right)}^2=\frac{4}{\pi }=\frac{4\times 7}{22}=\frac{14}{11}>1.$

So, Area of Circle $>$ Area of Square