A regular hexagon inscribed in a circle has side lengths equal to the radius of the circle. Therefore, the original side length \(s\) of the hexagon is equal to the radius of the circle, which is \(10\) cm. The perimeter \(P\) of a regular hexagon is given by:
["How the Side Length of a Regular Hexagon Inscibed in a Circle Relates to Its Radius (and Perimeter Calculations)", "A regular hexagon inscribed in a circle is a perfect example of geometric harmony, where symmetry and mathematical simplicity come together. Understanding this relationship not only reveals elegant properties of regular polygons but also simplifies calculations involving perimeter and circumradius.", "### The Unique Property: Side Length Equals Circle Radius", "If a regular hexagon is inscribed in a circle, the distance from the center of the circle to any vertex (the circle’s radius) is exactly equal to the length of each side of the hexagon. For instance, if the circle has a radius of (10) cm, then each side of the inscribed regular hexagon is also (10) cm.", "This geometric truth stems from the fact that the central angle subtended by each side is (60^\circ), forming six equilateral triangles. In every triangle, the two side lengths (radii) are equal, and so is the base — the hexagon side — proving that the side length (s) equals the radius (r) of the circumscribed circle.", "### The Perimeter of a Regular Hexagon", "The perimeter (P) of any regular polygon is calculated by multiplying the length of one side by the number of sides. For a regular hexagon with six equal sides:", "[\nP = 6 \ imes s\n]", "Since the side length (s = r) (equal to the radius), and given the radius is (10) cm:", "[\nP = 6 \ imes 10 = 60\ \ ext{cm}\n]", "Thus, the perimeter of a regular hexagon inscribed in a circle of radius (10) cm is (60) cm.", "### Why This Matters in Real Applications", "This clean relationship simplifies real-world calculations in architecture, design, and engineering. When planning circular layouts or hexagonal tile patterns, knowing that (s = r) allows rapid estimation of perimeters without complex formulas. It also serves as a foundational concept in trigonometry, polar coordinates, and tessellation theory.", "### Conclusion", "A regular hexagon inscribed in a circle demonstrates a pure mathematical relationship: the side length equals the circle’s radius. For a radius of (10) cm, each side is (10) cm, and the total perimeter is (60) cm. Recognizing this connection enables clearer geometric reasoning and efficient problem solving in both academic and practical settings.", "---", "Perimeter of a Regular Hexagon Formula:\n[\nP = 6 \ imes r\n]\nwhere (r) is the radius of the circumscribed circle. For (r = 10) cm,\n[\nP = 60\ \ ext{cm}.\n]"]
