Discussion
The area of a square is \(d\). What is the area of the circle which has the diagonal of the square as its diameter?
Answer: D
The area of the circle is \(\frac{1}{2}\pi d\).
Let the side of the square be \(s\), so \(s^{2} = d\). The diagonal is \(s\sqrt{2}\), and it is the diameter, so the radius is \(\frac{s\sqrt{2}}{2}\).
\[ \begin{aligned} \text{area} &= \pi \left(\frac{s\sqrt{2}}{2}\right)^{2} \\ &= \pi \cdot \frac{2s^{2}}{4} = \frac{1}{2}\pi d \end{aligned} \]
\(\pi d\) uses the diagonal as the radius, and \(\pi d^{2}\) and \(\frac{1}{4}\pi d^{2}\) treat \(d\) as a length rather than an area, so they have the wrong dimensions.
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