Question: A square with side length $ s $ is inscribed in a circle. If the circumference of the circle is $ 10\pi $ cm, what is the area of the square in square centimeters?

["A Square Inscribed in a Circle: Find the Area When Circumference is $10\pi$ cm", "Understanding the relationship between a square and the circle in which it’s inscribed unlocks key geometric insights useful in architecture, design, and advanced mathematics. In this article, we explore how to determine the area of a square whose sides measure $ s $, when it is perfectly inscribed in a circle with a circumference of $ 10\pi $ cm.", "### The Circle’s Circumference to Radius", "The circumference $ C $ of a circle is given by the formula:\n[ C = 2\pi r ]\nGiven $ C = 10\pi $ cm, we solve for the radius $ r $:\n[\n2\pi r = 10\pi \Rightarrow r = \frac{10\pi}{2\pi} = 5 \ ext{ cm}\n]", "### Radius and Diagonal of the Inscribed Square", "When a square is inscribed in a circle, the diagonal of the square equals the diameter of the circle. Since the radius is 5 cm, the diameter is:\n[ d = 2r = 10 \ ext{ cm} ]", "Let $ s $ be the side length of the square. The diagonal $ d $ of a square relates to its side length via the Pythagorean theorem:\n[ d = s\sqrt{2} ]\nSubstituting $ d = 10 $:\n[\ns\sqrt{2} = 10 \Rightarrow s = \frac{10}{\sqrt{2}} = 5\sqrt{2} \ ext{ cm}\n]", "### Area of the Square", "The area $ A $ of a square is $ s^2 $:\n[\nA = (5\sqrt{2})^2 = 25 \cdot 2 = 50 \ ext{ cm}^2\n]", "### Conclusion", "Thus, a square inscribed in a circle with circumference $ 10\pi $ cm has an area of 50 square centimeters. This elegant relationship between circles and inscribed polygons continues to be fundamental in geometry and real-world applications.", "If you're studying geometry or designing balanced shapes, remembering how a square’s diagonal aligns with a circle’s diameter can simplify many problems. Next time you encounter a such a configuration, use the diameter as the guide—your calculations will become faster and more intuitive."]