Question

If the circumference of a circle is $100\text{cm}$, then the side of a square inscribed in the circle is

• A. $5\sqrt{2}\text{cm}$
• B. $\frac{100}{\pi }\text{cm}$
• C. $\frac{50\sqrt{2}}{\pi }\text{cm}$
• D. $\frac{100\sqrt{2}}{\pi }\text{cm}$

Answer 1

The correct option is C $\frac{50\sqrt{2}}{\pi }\text{cm}$

Let the radius of circle be $r$ and side of square be $a$
Diagonal of square $=$ diameter of circle.
$\sqrt{a^2+a^2}=2r$
${\left(\sqrt{2a^2}\right)}^2=(2r{)}^2$
$2a^2=4r^2$
$\sqrt{a^2}=\sqrt{2r^2}$
$a=\pm \sqrt{2}r$
$a=\sqrt{2}r$ ... (1)
Circumference $=1000$
$2\pi r=100$
$r=\frac{100}{2\pi }$
$r=\frac{50}{\pi }$ ... (2)
Substituting $r=\frac{50}{\pi }$
From (2) in (1), we get:
$a=\sqrt{2}r$
$a=\sqrt{2}\times \frac{50}{\pi }$
$a=\frac{50\sqrt{2}}{\pi }\text{cm}$