眼見陳姑娘那條淺顯的問題，居然有咁多人都答不出。我出一題關於 2-polytope 的問題，讓各位思考一下....


(a) A regular 4-sided 2-polytope (4-gon) is inscribed in a circle with diameter 10. 4 smaller circles are then constructed with their diameters on each of the sides of the 2-polytope. The diagram is NOT drawn to scale. Find the shaded area.
(a)一個正4邊平面胞形, 即一個正四邊形, 鑲嵌在一個直徑有10 的圓形裡面。圓外有4個細圓另構成，鑲嵌的每條邊則當成其直徑。請找出已塗黑的面積。


(b) In general, the inscribed is a regular `n`-sided 2-polytope, or `n`-gon. Similarly, `n` smaller circles are constructed. Find the shaded area in general case.
(b) 若中間改為一個正`n`邊平面胞形, 即一個`n`多邊形, 鑲嵌在一個直徑有10 的圓形裡面。圓外有`n`個細圓另構成，鑲嵌的每條邊則當成其直徑。請找出其面積。