Question: A square with side length $ 8 $ cm is inscribed in a circle. What is the number of centimeters in the circumference of the circle? Express your answer in terms of $ \pi $.

["Question: A square with side length 8 cm is inscribed in a circle. What is the number of centimeters in the circumference of the circle?", "When a square is inscribed in a circle, the diagonal of the square becomes the diameter of the circle. This geometric relationship is key to solving the problem efficiently and accurately.", "Given that the side length of the square is $ 8 $ cm, we can calculate the length of the diagonal using the Pythagorean theorem. For a right triangle formed by two sides of the square and the diagonal:", "$$\n\ ext{Diagonal} = \sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} \ ext{ cm}\n$$", "Since the diagonal of the square is the diameter of the circle, the diameter $ d $ of the circle is $ 8\sqrt{2} $ cm. The circumference $ C $ of a circle is given by the formula:", "$$\nC = \pi d\n$$", "Substituting the diameter:", "$$\nC = \pi \ imes 8\sqrt{2} = 8\sqrt{2}\pi \ ext{ cm}\n$$", "Thus, the circumference of the circle is $ \boxed{8\sqrt{2}\pi} $ centimeters.", "This elegant solution highlights how understanding geometric properties—like the diagonal of an inscribed square—enables clean, accurate calculations directly tied to key formulas in geometry."]