Solution: In a regular hexagon, the side length is equal to the radius of the circumscribed circle. Thus, the radius $ r = 5 $ cm. The circumference $ C $ of the circle is given by $ C = 2\pi r $. Substituting, $ C = 2\pi \times 5 = 10\pi $.
["Understanding the Relationship Between Side Length and Circumradius in a Regular Hexagon", "When exploring geometric shapes, the regular hexagon stands out due to its symmetry and elegant mathematical properties. One fascinating insight is that in a regular hexagon, the side length is always equal to the radius of the circumscribed circle. This unique relationship simplifies many calculations and deepens our understanding of hexagonal geometry.", "If the side length of a regular hexagon is $ s = 5 $ cm, then the radius $ r $ of the circle that perfectly circumscribes the hexagon is also $ r = 5 $ cm. This equality stems from the hexagon’s structure—its six equal sides and angles allow each vertex to sit exactly on the circumference of a circle centered at the hexagon’s center.", "With $ r = 5 $ cm, the circumference $ C $ of the circumscribed circle can be calculated using the standard formula:", "[\nC = 2\pi r\n]", "Substituting the radius:", "[\nC = 2\pi \ imes 5 = 10\pi \ ext{ cm}\n]", "This result means the circumference of the circle is exactly $ 10\pi $ cm—a clean and compact expression that connects fundamental constants ($ \pi $) with precise geometric measurements.", "Why This Matters:", "- Efficiency in Design: Engineers and architects use these principles to create hexagonal structures that are structurally sound and material-efficient.\n- Mathematical Consistency: The equality $ s = r $ serves as a quick validation when verifying the fitness of a shape for hexagonal layouts.\n- Educational Value: Understanding this relationship helps students grasp the harmony between side length and circular boundaries in regular polygons.", "In summary, recognizing that the side length of a regular hexagon equals its circumradius (with $ s = r $) unlocks a powerful geometric insight. Combined with the simple yet profound formula $ C = 2\pi r $, this leads directly to $ C = 10\pi $ cm—showcasing how mathematical elegance supports practical applications in design, construction, and education.", "---", "Keywords: regular hexagon circumference, circumscribed circle radius, side length equals radius, circle circumference formula, 2πr, geometric relationships, hexagon geometry"]
