Question

What will be the area of the largest square that can be cut out of a circle of radius $10\mathrm{cm}$?

• A. $100{\mathrm{cm}}^2$
• B. $200{\mathrm{cm}}^2$
• C. $300{\mathrm{cm}}^2$
• D. $400{\mathrm{cm}}^2$

Answer 1

The correct option is B

$200{\mathrm{cm}}^2$

Explanation for the correct option.

The Square has a diagonal $\left(D\right)$ equal to the diameter of the circle.

Since the diameter of the circle is twice the radius.

Therefore,

$$\begin{gathered} \mathrm{Diameter}=2\times 10 \\ =20\mathrm{cm} \end{gathered}$$

Let us consider the side of the square be $a\mathrm{cm}$.

By using Pythagoras theorem,

$$\begin{gathered} D^2=a^2+a^2 \\ \Rightarrow {20}^2=2a^2 \\ \Rightarrow 400=2a^2 \\ \Rightarrow a^2=\frac{400}{2} \\ \Rightarrow a^2=200 \\ \Rightarrow a=\sqrt{200} \\ \therefore a=10\sqrt{2}\mathrm{cm} \end{gathered}$$

So, the area of the square is computed as,

$$\begin{gathered} \mathrm{Area}=\mathrm{side}\times \mathrm{side} \\ =10\sqrt{2}\times 10\sqrt{2} \\ =200{\mathrm{cm}}^2 \end{gathered}$$

Hence, option $B$ is correct.