Example Question - inscribed square

Area of a Circle with an Inscribed Square

<p>Let the side of the square be \( s \). Since the area of the square is 4, we have:</p> <p>\( s^2 = 4 \)</p> <p>\( s = 2 \)</p> <p>The diameter of the circle is the diagonal of the square. Using the Pythagorean theorem:</p> <p>\( \text{diagonal}^2 = s^2 + s^2 \)</p> <p>\( \text{diagonal}^2 = 2^2 + 2^2 \)</p> <p>\( \text{diagonal}^2 = 4 + 4 \)</p> <p>\( \text{diagonal}^2 = 8 \)</p> <p>\( \text{diagonal} = \sqrt{8} \)</p> <p>\( \text{diagonal} = 2\sqrt{2} \)</p> <p>Since the diameter is \( 2\sqrt{2} \), the radius \( r \) is \( \sqrt{2} \). The area \( A \) of the circle is:</p> <p>\( A = \pi r^2 \)</p> <p>\( A = \pi (\sqrt{2})^2 \)</p> <p>\( A = \pi \cdot 2 \)</p> <p>\( A = 2\pi \)</p> <p>Therefore, the correct answer is \( B: 2\pi \).</p>