# 圆的面积

## 算术证明

### 不大于

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

### 不小于

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

## 重排证明

${\displaystyle n}$                   面积
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
${\displaystyle \infty }$ ${\displaystyle {\frac {1}{\infty }}}$ ${\displaystyle \pi }$ 1 ${\displaystyle \pi }$

## 洋葱证明

{\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}}}

## 半圆证明

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

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

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

## 快速逼近

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

${\displaystyle k}$    ${\displaystyle n}$     ${\displaystyle u_{n}}$   ${\displaystyle U_{n}}$   ${\displaystyle {\frac {u_{n}+U_{n}}{4}}}$
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

### 推导

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

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

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

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

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

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

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

## 脚注

1. ^ 中文的“圆”可以指圆周（circle）也能指圆盘（disk），此文中“圆”指圆盘。

## 参考文献

• Archimedes, Measurement of a circle, T. L. Heath (trans.) (编), The Works of Archimedes, Dover: 91–93, 260 BCE, ISBN 978-0-486-42084-4
(Originally published by Cambridge University Press, 1897, based on J. L. Heiberg's Greek version.)
• Beckmann, Petr, A History of Pi, St. Martin's Griffin, 1976, ISBN 978-0-312-38185-1
• Gerretsen, J.; Verdenduin, P., Chapter 8: Polygons and Polyhedra, H. Behnke, F. Bachmann, K. Fladt, H. Kunle (eds.), S. H. Gould (trans.) (编), Fundamentals of Mathematics, Volume II: Geometry, MIT Press: 243–250, 1983, ISBN 978-0-262-52094-2
(Originally Grundzüge der Mathematik, Vandenhoeck & Ruprecht, Göttingen, 1971.)
• Laczkovich, Miklós, Equidecomposability and discrepancy: A solution to Tarski's circle squaring problem, Journal für die reine und angewandte Mathematik (Crelle’s Journal), 1990, 404: 77–117, ISSN 0075-4102[失效連結]
• Lange, Serge, The length of the circle, Math! : Encounters with High School Students, Springer-Verlag, 1985, ISBN 978-0-387-96129-3
• Smith, David Eugene; Mikami, Yoshio, A history of Japanese mathematics, Chicago: Open Court Publishing: 130–132, 1914, ISBN 978-0-87548-170-8
• Thijsse, J. M., Computational Physics, Cambridge University Press: p. 273, 2006, ISBN 978-0-5215-7588-1