The side length \( s \) of a regular hexagon inscribed in a circle is equal to the radius \( r \). For the original hexagon with radius 10 cm:

["The Fascinating Relationship: Side Length ( s ) of a Regular Hexagon Equals the Radius ( r ) in a Circumscribed Circle", "A regular hexagon inscribed in a circle is more than just a geometric pattern—it’s a perfect example of nature’s efficiency and mathematical elegance. One of the most intriguing properties is that the side length ( s ) of a regular hexagon perfectly matches the radius ( r ) of the circumscribed circle. This unique relationship simplifies many calculations in geometry and design. But what makes this true, and how does it apply when the radius is 10 cm?", "### Why Does the Side Length Equal the Radius?", "A regular hexagon can be divided into six equilateral triangles, each with vertices at the center of the circle and two adjacent vertices of the hexagon. Since all triangle sides are equal, and two of these sides are radii of the circle, the third side—the side of the hexagon—must also equal the radius. This geometric proof confirms why:", "[\ns = r\n]", "This simplicity reduces the complexity of calculations involving regular hexagons inscribed in circles.", "### Applying the Formula: Hexagon with Radius 10 cm", "If the radius ( r ) of the circle is 10 cm, then the side length ( s ) of the inscribed regular hexagon is:", "[\ns = r = 10,\ ext{cm}\n]", "This means each side of the hexagon measures exactly 10 cm, perfectly fitting the circle’s circumference.", "### Real-World Applications", "This relationship is used frequently in architecture, engineering, and natural pattern studies:", "- Architectural Tiling: Hexagonal flooring patterns often rely on this exact fit to minimize waste.\n- Nature Inspiration: Honeycombs and cellular structures mimic this shape due to its efficient space usage.\n- STEM Education: Demonstrating regular polygons and circles helps students grasp coordinate geometry and symmetry.", "### Summary", "For a regular hexagon inscribed in a circle:\n[\n\boxed{s = r}\n]\nWhen ( r = 10,\ ext{cm} ), then ( s = 10,\ ext{cm} ). This direct proportionality makes the regular hexagon a fascinating and practical shape in both theoretical math and real-world design.", "---", "Optimize your understanding and calculations of circular geometry—know that ( s = r ) is not just a rule but a powerful principle shaping efficient, aesthetically pleasing structures."]