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  "body": "\n## Definition\n\nLet $n \\geq 3$, and let $P_1 P_2 \\cdots P_n$ be a convex polygon in the [[Euclidean plane|0XW7V9ZZ05]] $\\mathbb{R}^2$. Write $P_0 = P_n$ and $P_{n+1} = P_1$.\nThe polygon is regular if all sides have the same length, $d(P_i, P_{i+1}) = d(P_1, P_2)$, and all [[angles|0GJAJC6TNZ]] are equal, $\\angle P_{i-1} P_i P_{i+1} = \\angle P_n P_1 P_2$, for $i = 1, \\dots, n$.\n\n## Remarks\n\nEach interior angle of a regular convex $n$-gon is $\\frac{(n - 2)\\pi}{n}$. Each exterior angle is $\\frac{2\\pi}{n}$.\nThe vertices are [[concyclic|049DCJZ448]], and the polygon also has an incircle. The vertices of a regular $n$-gon with the circumradius $1$ can be $\\left(\\cos \\frac{2\\pi k}{n}, \\sin \\frac{2\\pi k}{n}\\right)$, for $k = 0, \\dots, n - 1$.\nWith the side $s$, the area is $\\frac{1}{4} n s^2 \\cot \\frac{\\pi}{n}$.\nStar polygons, such as the pentagram, are also called regular polygons. They are not convex.\n\n## Explanations\n\n### L1 human\n\nA regular polygon looks the same from each corner. Equal sides alone are not enough: a rhombus has four equal sides but is regular only when it is a square. Equal angles alone are not enough either, as a rectangle shows.\n",
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