Discussion
A rhombus is formed by joining the midpoints of the sides of a unit square.
What is the diameter of the largest circle that can be inscribed within the rhombus?
Answer: A
The diameter is \(\frac{1}{\sqrt{2}}\).
Joining the midpoints of a unit square's sides gives a square (a rhombus with right angles) whose side is the distance between adjacent midpoints:
\[ \begin{aligned} s &= \sqrt{\left(\tfrac{1}{2}\right)^{2} + \left(\tfrac{1}{2}\right)^{2}} \\ &= \frac{1}{\sqrt{2}} \end{aligned} \]
The largest circle inside a square touches all four sides, so its diameter equals the side, \(1/\sqrt{2}\).
\(1/(2\sqrt{2})\) is the radius rather than the diameter. \(\sqrt{2}\) is the diagonal of the original square, and \(2\sqrt{2}\) is larger than the square itself.
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