A regular hexagon is inscribed in a circle with a radius of 12 cm. What is the perimeter of the hexagon?
["A Regular Hexagon Inscribed in a Circle: Understanding Its Perimeter", "When a regular hexagon is perfectly inscribed in a circle, its geometry reveals fascinating mathematical relationships. With a circle of radius 12 cm, this shape offers a perfect opportunity to explore symmetry, circle properties, and precise measurement. In this article, we’ll explain why a regular hexagon fits so elegantly in a circle and how to calculate its perimeter.", "---", "### Why a Regular Hexagon Fits in a Circle", "A regular hexagon has six equal sides and six equal internal angles of 120° each. When inscribed in a circle, each vertex touches the circle, and the distance from the center to any vertex is the radius—here, 12 cm. A key geometric fact is that in a circle of radius ( r ), a regular hexagon has sides that subtend a central angle of exactly ( 60^\circ ) (since ( 360^\circ \div 6 = 60^\circ )).", "This alignment ensures that each side of the hexagon is a chord of the circle subtending ( 60^\circ )—and because the central angle is equal to the radius’s field of symmetry, each side’s length equals the radius of the circle.", "---", "### How to Calculate the Side Length of the Hexagon", "Since each central angle is ( 60^\circ ), the triangle formed by two radii and one side of the hexagon is equilateral. In an equilateral triangle, all sides are equal. Therefore, the length of each side of the regular hexagon equals the radius of the circumscribed circle.", "[\n\ ext{Side length} = \ ext{Radius} = 12,\ ext{cm}\n]", "---", "### Calculating the Perimeter", "A regular hexagon has six identical sides. Thus, the perimeter ( P ) is simply:", "[\nP = 6 \ imes \ ext{side length} = 6 \ imes 12,\ ext{cm} = 72,\ ext{cm}\n]", "---", "### Summary", "- A regular hexagon inscribed in a circle uses the circle’s radius as the length of each of its six equal sides.\n- The central angle for each side is ( 60^\circ ), forming equilateral triangles.\n- With a radius of 12 cm, each side measures 12 cm.\n- Therefore, the perimeter is ( 6 \ imes 12 = 72,\ ext{cm} ).", "---", "### Final Answer", "[\n\boxed{72,\ ext{cm}}\n]", "Understanding this relationship helps in geometry, design, and even in real-world applications like architecture and engineering, where regular hexagons offer both aesthetic appeal and structural efficiency."]
