Question

Given circle \(O\) with a diameter of \(2\) and square \(ABCD\) inscribed within circle \(O\), what is the area of the shaded region (circle minus square)?

• A. \(2\)
• B. \(\pi - 2\)
• C. \(4\)
• D. \(4\pi - 2\)
• E.

Hint:

For a square inscribed in a circle, the square’s diagonal equals the circle’s diameter: \(s = \dfrac{\text{diameter}}{\sqrt{2}}\).

Answer 1

The Correct Option is B

Step 1: Area of the circle.
Diameter \(= 2 \Rightarrow r = 1\). Area \(= \pi r^2 = \pi\).
Step 2: Area of the inscribed square.
The diagonal of the square equals the circle’s diameter \(= 2\).
If \(s\) is the side, \(s\sqrt{2} = 2 \Rightarrow s = \sqrt{2}\).
Area of the square \(= s^2 = 2\).
Step 3: Shaded region.
\(\text{Area} = \pi - 2\).
Final Answer:
\[ \boxed{\pi - 2} \]