# 圓的面積

## 算術證明

### 不大於

{\displaystyle {\begin{aligned}E&{}=C-T\\&{}>G_{n}\\P_{n}&{}=C-G_{n}\\&{}>C-E\\P_{n}&{}>T\end{aligned}}}

### 不小於

{\displaystyle {\begin{aligned}D&{}=T-C\\&{}>G_{n}\\P_{n}&{}=C+G_{n}\\&{}

## 重排證明

n                   面積
4 1.4142136 2.8284271 0.7071068 2.0000000
6 1.0000000 3.0000000 0.8660254 2.5980762
8 0.7653669 3.0614675 0.9238795 2.8284271
10 0.6180340 3.0901699 0.9510565 2.9389263
12 0.5176381 3.1058285 0.9659258 3.0000000
14 0.4450419 3.1152931 0.9749279 3.0371862
16 0.3901806 3.1214452 0.9807853 3.0614675
96 0.0654382 3.1410320 0.9994646 3.1393502
1/∞ π 1 π

## 洋蔥證明

{\displaystyle {\begin{aligned}\mathrm {Area} (r)&{}=\int _{0}^{r}2\pi t\,dt\\&{}=\left[(2\pi ){\frac {t^{2}}{2}}\right]_{t=0}^{r}\\&{}=\pi r^{2}.\end{aligned}}}

## 半圓證明

${\displaystyle dx=r\cos \theta d\theta }$
${\displaystyle \theta =\arcsin \left({\frac {x}{r}}\right)}$

${\displaystyle =4\int _{0}^{r}{\sqrt {r^{2}-x^{2}}}\,dx}$
${\displaystyle =4\int _{0}^{\frac {\pi }{2}}{\sqrt {r^{2}(1-\sin ^{2}\theta )}}\cdot r\cos \theta \,d\theta }$
${\displaystyle =4r^{2}\int _{0}^{\frac {\pi }{2}}\cos ^{2}\theta \,d\theta }$

${\displaystyle =2r^{2}\int _{0}^{\frac {\pi }{2}}(1+\cos 2\theta )\,d\theta }$
${\displaystyle =2r^{2}\left[\theta +{\frac {1}{2}}\sin 2\theta \right]_{0}^{\frac {\pi }{2}}}$
${\displaystyle =\pi r^{2}.}$

## 快速逼近

${\displaystyle u_{2n}={\sqrt {U_{2n}u_{n}}}}$    （幾何平均
${\displaystyle U_{2n}={\frac {2U_{n}u_{n}}{U_{n}+u_{n}}}}$    （調和平均

k    n     un   Un   (un + Un)/4
0 6 6.0000000 6.9282032 3.2320508
1 12 6.2116571 6.4307806 3.1606094
2 24 6.2652572 6.3193199 3.1461443
3 48 6.2787004 6.2921724 3.1427182
4 96 6.2820639 6.2854292 3.1418733
5 192 6.2829049 6.2837461 3.1416628
6 384 6.2831152 6.2833255 3.1416102
7 768 6.2831678 6.2832204 3.1415970

${\displaystyle n{\frac {3\sin {\frac {\pi }{n}}}{2+\cos {\frac {\pi }{n}}}}<\pi

### 推導

{\displaystyle {\begin{aligned}c_{2n}^{2}&{}=\left(r+{\frac {1}{2}}c_{n}\right)2r\\c_{2n}&{}={\frac {s_{n}}{s_{2n}}}.\end{aligned}}}

${\displaystyle c_{2n}={\sqrt {2+c_{n}}}.\,\!}$

${\displaystyle c_{n}=2{\frac {s_{n}}{S_{n}}}.\,\!}$

${\displaystyle c_{2n}={\frac {s_{n}}{s_{2n}}}=2{\frac {s_{2n}}{S_{2n}}},}$

${\displaystyle u_{2n}^{2}=u_{n}U_{2n}.\,\!}$

${\displaystyle 2{\frac {s_{2n}}{S_{2n}}}{\frac {s_{n}}{s_{2n}}}=2+2{\frac {s_{n}}{S_{n}}},}$

${\displaystyle {\frac {2}{U_{2n}}}={\frac {1}{u_{n}}}+{\frac {1}{U_{n}}}.}$

## 腳註

1. ^ 中文的「圓」可以指圓周（circle）也能指圓盤（disk），此文中「圓」指圓盤。