A regular hexagon is inscribed in a circle with a radius of \(10\) cm. If each side of the hexagon is increased by \(2\) cm, what is the new perimeter of the hexagon? Express your answer in terms of the circle’s radius.

["Regular Hexagon Inscribed in a Circle: New Perimeter After Side Length Increase", "A regular hexagon inscribed in a circle offers a perfect blend of geometry and symmetry, making it a classic topic in mathematical education and design. When the radius of the circumscribed circle is known, the geometry of the hexagon becomes directly tied to that radius, facilitating precise calculations.", "In this article, we explore how adjusting the side length of such a hexagon affects its perimeter—and how this relates to the original circle’s radius—specifically when the radius is (10) cm and each side is increased by (2) cm.", "---", "### Relationship Between a Regular Hexagon and its Circumscribed Circle", "A key property of a regular hexagon inscribed in a circle is that its six sides are all equal in length, and each side is identical to the radius of the circle. This result stems from the geometry of equilateral triangles formed by the center and adjacent vertices of the hexagon.", "Thus, when the circle has a radius (r = 10) cm, each side of the regular inscribed hexagon is:", "[\ns = r = 10 \ ext{ cm}\n]", "Since a hexagon has 6 sides, its original perimeter is:", "[\nP = 6 \ imes r = 6 \ imes 10 = 60 \ ext{ cm}\n]", "---", "### Increasing Each Side by 2 cm", "Now, each side is increased by (2) cm:", "[\ns_{\ ext{new}} = 10 + 2 = 12 \ ext{ cm}\n]", "With all six sides now equal in length, the new perimeter becomes:", "[\nP_{\ ext{new}} = 6 \ imes 12 = 72 \ ext{ cm}\n]", "Alternatively, expressing the result in terms of the original radius (r = 10) cm:", "Since (s_{\ ext{new}} = r + 2), then:", "[\nP_{\ ext{new}} = 6(r + 2) = 6r + 12\n]", "Substituting (r = 10):", "[\nP_{\ ext{new}} = 6(10) + 12 = 72 \ ext{ cm}\n]", "---", "### Conclusion", "When a regular hexagon is inscribed in a circle of radius (10) cm, each side measures (10) cm, giving a perimeter of (60) cm. After increasing each side by (2) cm, the new perimeter is (72) cm—equivalent to (6(r + 2)), which expresses the updated perimeter in terms of the original circle radius.", "This example highlights how geometric symmetries and side-length relationships enable clear and scalable computations in regular polygons inscribed in circles.", "---", "Summary:\nNew perimeter = (6(r + 2)) cm, where (r = 10) cm.\nFinal answer: (72) cm, consistent with (6(10 + 2) = 72)."]