A 5 cm by 12 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 Relationship: A 5 cm by 12 cm Rectangle Inscribed in a Circle", "When a rectangle is inscribed in a circle, the circle’s diameter equals the rectangle’s diagonal. This crucial insight allows us to calculate the circle’s circumference using basic geometry. In this article, we’ll explore how to determine the circumference of the circle that perfectly fits a 5 cm by 12 cm rectangle.", "### Step 1: Identify the Key Relationship\nSince the rectangle is inscribed in the circle, the rectangle’s two opposite corners touch the circle’s boundary, forming the diagonal as the diameter of the circle. Therefore, the diagonal of the rectangle serves as the circle’s diameter.", "### Step 2: Use the Pythagorean Theorem\nTo find the length of the diagonal, apply the Pythagorean theorem. For a rectangle with sides 5 cm and 12 cm, the diagonal (d) is:", "[\nd = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \ ext{ cm}\n]", "Thus, the diameter of the circle is 13 cm.", "### Step 3: Calculate the Circumference\nThe circumference (C) of a circle is given by the formula:", "[\nC = \pi \ imes \ ext{diameter}\n]", "Substituting the diameter:", "[\nC = \pi \ imes 13 = 13\pi \ ext{ cm}\n]", "### Conclusion\nThe circumference of the circle in which a 5 cm by 12 cm rectangle is inscribed is (13\pi) centimeters—expressed neatly in terms of (\pi). This elegant result arises from the geometric relationship between rectangles and their circumscribed circles, proving how fundamental shapes can simplify even complex calculations.", "Whether you’re solving geometry problems or designing circular structures, understanding inscribed rectangles and their diameters is essential for accurate measurements and efficient planning.", "---", "Key Takeaways:\n- A rectangle inscribed in a circle has the circle’s diameter equal to the rectangle’s diagonal.\n- Use the Pythagorean theorem to compute the diagonal: (d = \sqrt{a^2 + b^2}).\n- Circumference = (\pi \ imes \ ext{diameter}).\n- For a 5 cm × 12 cm rectangle, circumference = (13\pi) cm."]
