5Question: A historian of science is studying an ancient diagram of a 10 cm by 24 cm rectangle inscribed in a circle. What is the circumference of the circle, in centimeters, expressed in terms of $\pi$?
["Understanding the Geometry: Finding the Circumference of a Circle Circumscribing a Rectangle", "When studying ancient scientific diagrams, one often encounters elegant geometric relationships that bridge mathematics and artistry. A particularly insightful example involves a 10 cm by 24 cm rectangle inscribed in a circle—a configuration that reveals deep connections between rectangles, circles, and the life cycles of scientific inquiry. For historians of science analyzing such diagrams, understanding how to compute the circle’s circumference based on the rectangle’s dimensions offers both mathematical rigor and historical context.", "In this famous geometric arrangement, the rectangle is perfectly inscribed in the circle, meaning all four corners of the rectangle touch the circle’s circumference. This fact provides a powerful insight: the diagonal of the rectangle is the diameter of the circumscribed circle.", "To find the circumference, we begin by calculating the length of the diagonal using the Pythagorean Theorem:", "[\nd = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26~\ ext{cm}\n]", "Since the diagonal equals the diameter ($d = 26$ cm), the radius $r$ of the circle is:", "[\nr = \frac{d}{2} = \frac{26}{2} = 13~\ ext{cm}\n]", "The circumference $C$ of a circle is given by the formula:", "[\nC = 2\pi r\n]", "Substituting $r = 13$ cm:", "[\nC = 2\pi \ imes 13 = 26\pi~\ ext{cm}\n]", "Thus, the circumference of the circle is $26\pi$ centimeters—a result deeply consistent with ancient geometric reasoning. The ratio of 5:24:26, derived from the 5–12–13 Pythagorean triple (scaled by 2), reflects the meticulous precision valued by early mathematicians.", "For historians of science, this diagram is more than a geometric curiosity—it illustrates how ancient scholars intuitively grasped the harmony between straight lines, symmetry, and circular motion. By analyzing such sketches, we uncover not just equations, but the intellectual spirit behind early scientific exploration.", "In summary, the circumference of the circle inscribing a 10 cm by 24 cm rectangle is $\boxed{26\pi}$ centimeters."]
