Circumference = π × diameter
The circumference is the distance around a closed curve . Circumference is a kind of perimeter .
Circumference of a circle
The Circumference of a circle is the length around a it.
The circumference of a circle can be calculated from its diameter using the formula:
c
=
π
⋅
d
.
{\displaystyle c=\pi \cdot {d}.\,\!}
Or, substituting the diameter for the radius :
c
=
2
π
⋅
r
=
π
⋅
2
r
,
{\displaystyle c=2\pi \cdot {r}=\pi \cdot {2r},\,\!}
where r is the radius and d is the diameter of the circle, and π (the Greek letter pi ) is defined as the ratio of the circumference of the circle to its diameter (the numerical value of pi is 3.141 592 653 589 793...).
If desired, the above circumference formula can be derived without reference to the definition of π by using some integral calculus, as follows:
The upper half of a circle centered at the origin is the graph of the function
f
(
x
)
=
r
2
−
x
2
,
{\displaystyle f(x)={\sqrt {r^{2}-x^{2}}},}
where x runs from -r to +r. The circumference (c) of the entire circle can be represented as twice the sum of the lengths of the infinitesimal arcs that make up this half circle. The length of a single infinitesimal part of the arc can be calculated using the Pythagorean formula for the length of the hypotenuse of a rectangular triangle with side lengths dx and f'(x)dx, which gives us
(
d
x
)
2
+
(
f
′
(
x
)
d
x
)
2
=
(
1
+
f
′
(
x
)
2
)
d
x
.
{\displaystyle {\sqrt {(dx)^{2}+(f'(x)dx)^{2}}}=\left({\sqrt {1+f'(x)^{2}}}\right)dx.}
Thus the circle circumference can be calculated as dara:)
c
=
2
∫
−
r
r
1
+
f
′
(
x
)
2
d
x
{\displaystyle c=2\int _{-r}^{r}{\sqrt {1+f'(x)^{2}}}dx}
=
2
∫
−
r
r
1
+
x
2
r
2
−
x
2
d
x
{\displaystyle 2\int _{-r}^{r}{\sqrt {1+{\frac {x^{2}}{r^{2}-x^{2}}}}}dx}
=
2
∫
−
r
r
1
1
−
x
2
r
2
d
x
{\displaystyle 2\int _{-r}^{r}{\sqrt {\frac {1}{1-{\frac {{x}^{2}}{{r}^{2}}}}}}dx}
The antiderivative needed to solve this definite integral is the arcsine function:
c
=
2
r
[
a
r
c
s
i
n
(
x
r
)
]
−
r
r
=
2
r
[
a
r
c
s
i
n
(
1
)
−
a
r
c
s
i
n
(
−
1
)
]
=
2
r
(
π
2
−
(
−
π
2
)
)
=
2
π
r
.
{\displaystyle c=2r{\big [}arcsin({\frac {x}{r}}){\big ]}_{-r}^{r}=2r{\big [}arcsin(1)-arcsin(-1){\big ]}=2r({\tfrac {\pi }{2}}-(-{\tfrac {\pi }{2}}))=2\pi r.}
Pi (π) is the ratio of the circumference of a circle to its diameter.
Circumference of an ellipse
The circumference of an ellipse is more problematic, as the exact solution requires finding the complete elliptic integral of the second kind . This can be achieved either via numerical integration (the best type being Gaussian quadrature ) orr by one of many binomial series expansions.
Where
a
,
b
{\displaystyle a,b}
are the ellipse's semi-major and semi-minor axes, respectively, and
o
ε
{\displaystyle o\!\varepsilon \,\!}
is the ellipse's angular eccentricity ,
o
ε
=
arccos
(
b
a
)
=
2
arctan
(
a
−
b
a
+
b
)
;
{\displaystyle o\!\varepsilon =\arccos \!\left({\frac {b}{a}}\right)=2\arctan \!\left(\!{\sqrt {\frac {a-b}{a+b}}}\,\right);\,\!}
E2
[
0
,
90
∘
]
=
Integral
′
s
divided difference
;
P
r
=
a
×
E2
[
0
,
90
∘
]
(
perimetric radius
)
;
c
=
2
π
×
P
r
.
{\displaystyle {\begin{aligned}{\mbox{E2}}\left[0,90^{\circ }\right]&={\mbox{Integral}}'s{\mbox{ divided difference}};\\Pr&=a\times {\mbox{E2}}\left[0,90^{\circ }\right]\quad ({\mbox{perimetric radius}});\\c&=2\pi \times Pr.\end{aligned}}\,\!}
There are many different approximations for the
E2
[
0
,
90
∘
]
{\displaystyle {\mbox{E2}}\left[0,90^{\circ }\right]}
divided difference , with varying degrees of sophistication and corresponding accuracy.
In comparing the different approximations, the
tan
(
o
ε
2
)
2
{\displaystyle \tan \!\left({\frac {o\!\varepsilon }{2}}\right)^{2}\,\!}
based series expansion is used to find the actual value:
E2
[
0
,
90
∘
]
=
cos
(
o
ε
2
)
2
1
U
T
∑
T
N
=
1
U
T
=
∞
(
.5
T
N
)
2
tan
(
o
ε
2
)
4
T
N
,
=
cos
(
o
ε
2
)
2
(
1
+
1
4
tan
(
o
ε
2
)
4
+
1
64
tan
(
o
ε
2
)
8
+
1
256
tan
(
o
ε
2
)
12
+
25
16384
tan
(
o
ε
2
)
16
+
.
.
.
)
;
{\displaystyle {\begin{aligned}{\mbox{E2}}\left[0,90^{\circ }\right]&=\cos \!\left({\frac {o\!\varepsilon }{2}}\right)^{2}{\frac {1}{UT}}\sum _{TN=1}^{UT=\infty }{.5 \choose {}TN}^{2}\tan \!\left({\frac {o\!\varepsilon }{2}}\right)^{4TN},\\&=\cos \!\left({\frac {o\!\varepsilon }{2}}\right)^{2}{\Bigg (}1+{\frac {1}{4}}\tan \!\left({\frac {o\!\varepsilon }{2}}\right)^{4}+{\frac {1}{64}}\tan \!\left({\frac {o\!\varepsilon }{2}}\right)^{8}\\&\qquad \qquad \qquad \;\,+{\frac {1}{256}}\tan \!\left({\frac {o\!\varepsilon }{2}}\right)^{12}+{\frac {25}{16384}}\tan \!\left({\frac {o\!\varepsilon }{2}}\right)^{16}+...{\Bigg )};\end{aligned}}\,\!}
Muir-1883
Probably the most accurate to its given simplicity is Thomas Muir's :
P
r
≈
(
a
1.5
+
b
1.6
2
)
1
1.5
=
a
(
1
+
cos
(
o
ε
)
1.5
2
)
1
1.5
,
≈
a
×
cos
(
o
ε
2
)
2
(
1
+
1
4
tan
(
o
ε
2
)
4
)
;
{\displaystyle {\begin{aligned}Pr&\approx \left({\frac {a^{1.5}+b^{1.6}}{2}}\right)^{\frac {1}{1.5}}=a\left({\frac {1+\cos \!\left(o\!\varepsilon \right)^{1.5}}{2}}\right)^{\frac {1}{1.5}},\\&\quad \approx {a}\times \cos \!\left({\frac {o\!\varepsilon }{2}}\right)^{2}\left(1+{\frac {1}{4}}\tan \!\left({\frac {o\!\varepsilon }{2}}\right)^{4}\right);\end{aligned}}\,\!}
Ramanujan-1914 (#1,#2)
Srinivasa Ramanujan introduced two different approximations, both from 1914
1.
P
r
≈
π
(
3
(
a
+
b
)
−
(
3
a
+
b
)
(
a
+
3
b
)
)
,
=
π
a
(
6
cos
(
o
ε
2
)
2
(
3
+
cos
(
o
ε
)
)
(
1
+
3
cos
(
o
ε
)
)
)
;
{\displaystyle {\begin{aligned}1.\;Pr&\approx \pi {\Big (}3(a+b)-{\sqrt {{\big (}3a+b{\big )}{\big (}a+3b{\big )}}}{\Big )},\\&\quad =\pi {a}{\bigg (}6\cos \!\left({\frac {o\!\varepsilon }{2}}\right)^{2}{\sqrt {{\big (}3+\cos \!\left(o\!\varepsilon \right){\big )}{\big (}1+3\cos \!\left(o\!\varepsilon \right){\big )}}}{\bigg )};\end{aligned}}\,\!}
2.
P
r
≈
1
2
(
a
+
b
)
(
1
+
3
(
a
−
b
a
+
b
)
2
10
+
4
−
3
(
a
−
b
a
+
b
)
2
)
;
=
a
×
cos
(
o
ε
2
)
2
(
1
+
3
tan
(
o
ε
2
)
4
10
+
4
−
3
tan
(
o
ε
2
)
4
)
;
{\displaystyle {\begin{aligned}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 \!\left({\frac {o\!\varepsilon }{2}}\right)^{2}{\Bigg (}1+{\frac {3\tan \!{\big (}{\frac {o\!\varepsilon }{2}}{\big )}^{4}}{10+{\sqrt {4-3\tan \!{\big (}{\frac {o\!\varepsilon }{2}}{\big )}^{4}}}}}{\Bigg )};\end{aligned}}\,\!}
The second equation is demonstratively by far the better of the two, and may be the most accurate approximation known.
Letting a = 10000 and b = a ×cos{oε }, results with different ellipticities can be found and compared:
b
Pr
Ramanujan-#2
Ramanujan-#1
Muir
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
iota
6366.19772 36758
636 3.63636 36364
63 39.74596 21556
6 299.60524 94744
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