question_answer

If a square is inscribed in a circle, what is the ratio of the areas of the circle and the square?

A) \[2:\pi \]

B) \[\pi :2\]

C) \[\pi :4\]

D) \[4:\pi \]

Correct Answer: B

Solution :

[b] Let the side of square be 'a' units and the radius of circle be 'r' units.

In right-angled \[\Delta ABC,\]

\[A{{C}^{2}}=B{{C}^{2}}+A{{B}^{2}}\]

(By Pythagoras theorem)

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,{{(2r)}^{2}}={{a}^{2}}+{{a}^{2}}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,4{{r}^{2}}=2{{a}^{2}}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,2{{r}^{2}}={{a}^{2}}\]

Now,     \[\frac{\text{Area of circle}}{\text{Area of square}}=\frac{\pi {{r}^{2}}}{{{a}^{2}}}=\frac{\pi {{r}^{2}}}{2{{r}^{2}}}=\frac{\pi }{2}\]

Hence, the required ratio is \[\pi :2\].