Question: A 5 cm by 8 cm rectangle is inscribed in a circle. What is the circumference of the circle in centimeters? Express your answer in terms of $ \pi $.
["Understanding the Circumference of a Circle Through a Carved Rectangle: A 5 cm by 8 cm Rectangle in a Circle", "When a rectangle is inscribed in a circle, the circle’s diameter is precisely equal to the diagonal of the rectangle. This geometric relationship provides a straightforward way to calculate the circle’s circumference — a classic example of how real-world shapes connect to fundamental mathematical principles.", "Let’s explore how to determine the circumference of the circle that perfectly encloses a 5 cm by 8 cm rectangle.", "### Step 1: Diagonal of the Rectangle as the Diameter", "For a rectangle inscribed in a circle, the rectangle’s diagonals are equal and correspond to the diameter of the circle. We can find the length of the diagonal using the Pythagorean theorem:", "[\nd = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89} , \ ext{cm}\n]", "Thus, the diameter of the circle is $ \sqrt{89} $ cm.", "### Step 2: Calculate the Circumference", "The circumference $ C $ of a circle is given by the formula:", "[\nC = \pi \ imes \ ext{diameter}\n]", "Substituting the diameter:", "[\nC = \pi \ imes \sqrt{89} = \sqrt{89}\pi , \ ext{cm}\n]", "### Final Answer", "The circumference of the circle in which a 5 cm by 8 cm rectangle is inscribed is:", "[\n\sqrt{89}\pi , \ ext{cm}\n]", "### Why This Matters", "This problem illustrates a key concept in geometry — the relationship between inscribed shapes and their circumscribing circles. It’s not just about finding a number — it’s about recognizing how geometric form and dimension interact. Whether you're designing a circular frame or solving a construction problem, such principles empower precise calculations.", "In summary:\nA 5 cm by 8 cm rectangle inscribed in a circle has a circumference of $ \sqrt{89}\pi $ centimeters — a clean, precise expression in terms of $ \pi $, rooted in the diagonal as the diameter."]
