Question

Question 3
If the circumference of a circle and the perimeter of a square are equal, then:
(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

According to the given condition,

Circumference of a circle = Perimeter of square

$2\pi R=4a$

[ Where r and a are radius of circle and side of square respectively ]

$\Rightarrow \frac{22}{7}r=2a\Rightarrow 11r=7a$

$\Rightarrow a-\frac{11}{7}r\Rightarrow r=\frac{7a}{11}$

Now, area of circle $A_1=\pi r^2$

$=\pi {\left(\frac{7a}{11}\right)}^2=\frac{22}{7}\times \frac{49a^2}{121}$

$=\frac{14a^2}{11}$

And area of square, $A_2=(a{)}^2$

From Eqs (ii) and (iii), $A_1=\frac{14}{11}{A}_2$

$A_1>A_2$

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