Solution: To find the circumference of the circle in which a rectangle is inscribed, we first recognize that the diagonal of the rectangle is the diameter of the circle. Using the Pythagorean theorem, the diagonal $d$ of a 5 cm by 8 cm rectangle is:
["Finding the Circumference of a Circle That Inscribes a Rectangle: A Step-by-Step Solution", "When tasked with finding the circumference of a circle in which a rectangle is inscribed, the key insight lies in geometry: the diagonal of the rectangle serves as the diameter of the circle. This elegant relationship allows us to use the Pythagorean theorem to compute the diagonal and, subsequently, the circumference.", "In this article, we explore how to determine the circumference of the circumscribed circle using a specific example — a rectangle measuring 5 cm by 8 cm — while clearly explaining the mathematical principles involved.", "---", "### Why the Diagonal of the Rectangle Is the Diameter", "A rectangle inscribed in a circle means all four vertices of the rectangle touch the circle’s boundary. The only way this is possible is if the rectangle’s diagonal passes through the center of the circle and spans the diameter. This geometric property ensures that the diagonal connects two opposite points on the circle’s circumference, confirming it as the diameter.", "---", "### Applying the Pythagorean Theorem", "To compute the diagonal $ d $ of a rectangle with side lengths $ a = 5 , \ ext{cm} $ and $ b = 8 , \ ext{cm} $, we use the Pythagorean theorem:", "[\nd = \sqrt{a^2 + b^2} = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89} , \ ext{cm}\n]", "So, the diameter of the circle is $ \sqrt{89} , \ ext{cm} $.", "---", "### Calculating the Circumference", "With the diameter $ d = \sqrt{89} , \ ext{cm} $, the circumference $ C $ of the circle is given by the formula:", "[\nC = \pi \ imes d = \pi \ imes \sqrt{89} , \ ext{cm}\n]", "This exact expression is ideal for precise applications, but if a decimal approximation is preferred:", "[\nC \approx \pi \ imes 9.434 \approx 29.69 , \ ext{cm}\n]", "---", "### Final Result Summary", "- Rectangle dimensions: 5 cm × 8 cm\n- Diagonal (diameter): $ \sqrt{89} , \ ext{cm} \approx 9.434 , \ ext{cm} $\n- Circumference: $ C = \pi \sqrt{89} , \ ext{cm} \approx 29.69 , \ ext{cm} $", "---", "### Why This Method Matters", "Finding the circumference in an inscribed rectangle is a classic application of the Pythagorean theorem in practical geometry. Understanding that the diagonal becomes the diameter unlocks solutions for numerous problems involving circles, architecture, construction, and design. Whether you're a student mastering geometry or a professional solving spatial problems, this solution provides a clear, accurate pathway.", "---", "In summary: Recognize the rectangle diagonal as the circle’s diameter, apply the Pythagorean theorem to find this diagonal, then compute the circumference using $ C = \pi d $. This solution applies universally to any rectangle inscribed in a circle — not just 5 cm by 8 cm.", "---", "Keywords: circumference of circle, inscribed rectangle, diagonal as diameter, Pythagorean theorem, find circle diameter, geometry solution, circular geometry, 5x8 rectangle diagonal, circular perimeter, circle from inscribed rectangle", "Meta Description: Learn how to find the circumference of a circle that inscribes a rectangle using the Pythagorean theorem. Step-by-step with example: rectangle 5 cm by 8 cm → diagonal is diameter → circumference = π√89 cm ≈ 29.69 cm."]
