원둘레: 두 판 사이의 차이
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5번째 줄:
== 공식 ==
: <math>c=\pi\cdot{d}.\,\!</math>
== 타원의 둘레 ==
=== 뮤어 1883 ===
[[토머스 뮤어]] 공식:
::<math>\begin{align}Pr
&\approx\left(\frac{a^{1.5}+b^{1.5}}{2}\right)^\frac{1}{1.5}=a\left(\frac{2+\cos^{3}\!\left(\alpha\right)}{2}\right)^\frac{2}{3},\\
&\quad\approx{a}\times\cos^2\!\left(\frac{\alpha}{2}\right)\left(1+\frac{1}{4}\tan^4\!\left(\frac{\alpha}{2}\right)\right).\end{align}\,\!</math>
=== 라마누잔-1914 (#1,#2) ===
[[스리니바사 라마누잔]]의 공식:
::<math>\begin{align}1.\;Pr&\approx\frac{1}{2}\Big(3(a+b)-\sqrt{\big(3a+b\big)\big(a+3b\big)}\Big),\\
&\quad=\frac{a}{2}\bigg(6\cos^2\!\left(\frac{\alpha}{2}\right)-\sqrt{\big(3+\cos\!\left(\alpha\right)\big)\big(1+3\cos\!\left(\alpha\right)\big)}\bigg);\end{align}\,\!</math>
::<math>\begin{align}2.\;Pr&\approx\frac{1}{2}\Big(a+b\Big)\Bigg(1+\frac{3\big(\frac{a-b}{a+b}\big)^2}{10+\sqrt{4-3\big(\frac{a-b}{a+b}\big)^2}}\Bigg);\\
&\quad=a\times\cos^2\!\left(\frac{\alpha}{2}\right)\Bigg(1+\frac{3\tan^4\!\big(\frac{\alpha}{2}\big)}{10+\sqrt{4-3\tan^4\!\big(\frac{\alpha}{2}\big)}}\Bigg).\end{align}\,\!</math>
a = 10000, b = a cos{oε}:
{|class="wikitable" HII
|-
! b !! Pr !! 라마누잔-#2 !! 라마누잔-#1 !! 뮤어
|-
|9975
|| '''9987.50391 11393'''
|| '''9987.50391 11393'''
|| '''9987.50391 11393'''
|| '''9987.50391 113'''89
|-
|9966
|| '''9983.00723 73047'''
|| '''9983.00723 73047'''
|| '''9983.00723 73047'''
|| '''9983.00723 730'''34
|-
|9950
|| '''9975.01566 41666'''
|| '''9975.01566 41666'''
|| '''9975.01566 41666'''
|| '''9975.01566 416'''04
|-
|9900
|| '''9950.06281 41695'''
|| '''9950.06281 41695'''
|| '''9950.06281 41695'''
|| '''9950.06281 4'''0704
|-
|9000
|| '''9506.58008 71725'''
|| '''9506.58008 71725'''
|| '''9506.58008''' 67774
|| '''9506.5'''7894 84209
|-
|8000
|| '''9027.79927 77219'''
|| '''9027.79927 77219'''
|| '''9027.7992'''4 43886
|| '''9027.7'''7786 62561
|-
|7500
|| '''8794.70009 24247'''
|| '''8794.70009 2424'''0
|| '''8794'''.69994 52888
|| '''8794'''.64324 65132
|-
|6667
|| '''8417.02535 37669'''
|| '''8417.02535 37'''460
|| '''8417.02'''428 62059
|| '''841'''6.81780 56370
|-
|5000
|| '''7709.82212 59502'''
|| '''7709.82212''' 24348
|| '''7709.8'''0054 22510
|| '''770'''8.38853 77837
|-
|3333
|| '''7090.18347 61693'''
|| '''7090.183'''24 21686
|| '''70'''89.94281 35586
|| '''70'''83.80287 96714
|-
|2500
|| '''6826.49114 72168'''
|| '''6826.4'''8944 11189
|| '''682'''5.75998 22882
|| '''68'''14.20222 31205
|-
|1000
|| '''6468.01579 36089'''
|| '''646'''7.94103 84016
|| '''646'''2.57005 00576
|| '''64'''31.72229 28418
|-
| 100
|| '''6367.94576 97209'''
|| '''636'''6.42397 74408
|| '''63'''46.16560 81001
|| '''63'''03.80428 66621
|-
| 10
|| '''6366.22253 29150'''
|| '''636'''3.81341 42880
|| '''63'''40.31989 06242
|| '''6'''299.73805 61141
|-
| 1
|| '''6366.19804 50617'''
|| '''636'''3.65301 06191
|| '''63'''39.80266 34498
|| '''6'''299.60944 92105
|-
|[[:en:iota|iota]]
|| '''6366.19772 36758'''
|| '''636'''3.63636 36364
|| '''63'''39.74596 21556
|| '''6'''299.60524 94744
|}
== 원주각 ==
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