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) none of these
(b) Area of the circle > Area of the square
Let r be the radius of the circle.
We know:
Circumference of the circle $=2 \pi r$
Now,
Let a be the side of the square.
We know:
Perimeter of the square = 4a
Now,
$2 \pi r=4 a$
$\Rightarrow r=\frac{4 a}{2 \pi}$
$\therefore$ Area of the circle $=\pi r^{2}$
$=\pi \times\left(\frac{4 a}{2 \pi}\right)^{2}$
$=\pi \times \frac{16 a^{2}}{4 \pi^{2}}$
$=\frac{4 a^{2}}{\pi}$
$=\frac{4 \times 7 a^{2}}{22}$
$=\frac{14 a^{2}}{11}$
Also,
Area of the square $=a^{2}$
Clearly, $\frac{14 a^{2}}{11}>a^{2}$
∴ Area of the circle > Area of the square
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