单位圆周上任取n个点, 构成的n边形面积是多少

northwolves

王守恩 发表于 2026-5-7 08:39
根据6#资料——是这串数？？？

{0.47746483, 0.95492966, 1.3496629, 1.6616646, 1.9063846, 2.0992728,  ...

1. Table[{n,\[Pi] HypergeometricPFQ[{1},{(n+1)/2,(n+2)/2},-\[Pi]^2]//FullSimplify},{n,3,10}]

复制代码

$
\begin{array}{cc}
3 & \frac{3}{2 \pi } \\
4 & \frac{3}{\pi } \\
5 & \frac{5 \left(2 \pi ^2-3\right)}{2 \pi ^3} \\
6 & \frac{15 \left(\pi ^2-3\right)}{2 \pi ^3} \\
7 & \frac{21 \left(2 \pi ^2 \left(\pi ^2-5\right)+15\right)}{4 \pi ^5} \\
8 & \frac{7 \left(45-15 \pi ^2+2 \pi ^4\right)}{\pi ^5} \\
9 & \frac{9 \left(-315+210 \pi ^2-42 \pi ^4+4 \pi ^6\right)}{2 \pi ^7} \\
10 & \frac{45 \left(-315+105 \pi ^2-14 \pi ^4+\pi ^6\right)}{2 \pi ^7} \\
\end{array}
$

王守恩

备忘——圆周上任取 n 个点,  求 n 个点落在同一半圆周内的概率。

Table[{n, n/2^(n - 1)}, {n, 10}]

{{1, 1}, {2, 1}, {3, 3/4}, {4, 1/2}, {5, 5/16}, {6, 3/16}, {7, 7/64}, {8, 1/16}, {9, 9/256}, {10, 5/256}}