Solution: The diagonal of the rectangle is the diameter of the circumscribed circle. Using the Pythagorean theorem, the diagonal $ d = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26 $ km. The circumference $ C = \pi \times d = 26\pi $ km. Thus, the circumference is $ \boxed{26\pi} $.
["Solution: The Diagonal of a Rectangle Is the Diameter of Its Circumscribed Circle", "Understanding geometric relationships is fundamental in mathematics, especially when solving real-world problems involving shapes and measurements. One important principle is that the diagonal of a rectangle always serves as the diameter of its circumscribed circle — a concept that simplifies many calculations in geometry and design.", "In this article, we’ll explore how applying the Pythagorean theorem to find the diagonal of a rectangle enables us to calculate the circumference of the circle that perfectly fits around it. This solution not only demonstrates mathematical elegance but also provides practical utility in fields like architecture, engineering, and cartography.", "---", "### Step 1: The Rectangle’s Diagonal as the Circle’s Diameter", "Consider a rectangle with side lengths 10 km and 24 km. To find the circle around which this rectangle is inscribed — its circumscribed circle — the key insight is recognizing that the diagonal stretches across the rectangle’s center, touching all four corners, making it the circle’s diameter.", "### Step 2: Calculate the Diagonal Using the Pythagorean Theorem", "Using Pythagoras’ theorem, the diagonal $ d $ satisfies:\n[\nd = \sqrt{a^2 + b^2}\n]\nwhere $ a = 10 $ km and $ b = 24 $ km.", "Substitute the values:\n[\nd = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26 \ ext{ km}\n]", "Thus, the diameter of the circumscribed circle is 26 km.", "### Step 3: Compute the Circumference of the Circle", "The circumference $ C $ of a circle is given by $ C = \pi \ imes \ ext{diameter} $. Substituting $ d = 26 $ km:\n[\nC = \pi \ imes 26 = 26\pi \ ext{ km}\n]", "---", "### Final Result", "The circumference of the circumscribed circle is:\n[\n\boxed{26\pi} \ ext{ km}\n]", "---", "This elegant geometric solution proves how foundational theorems like the Pythagorean theorem enable precise calculations, bridging theory and real-world application. Whether designing circular paths, marking land boundaries, or modeling structural integrity, recognizing the diagonal as the circle’s diameter ensures accurate and efficient results."]
