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The NUMBER PI

There are many forms of PI and you can see them

However the number of Pi is not just a circle but a series of geometric images such as an equilateral triangle, square, pentagon, etc. unique pattern that I found 2004. Pi for all numbers is:

π n = n 2 . s i n ( 360 n ) {\displaystyle \pi _{n}={\frac {n}{2}}.sin({\frac {360}{n}})}

sinus in degrees

n= {3,4,5,6........., infinity}
n=3 it is the equilateral triangle
π 3 = 3 2 sin ⁡ ( 360 3 ) {\displaystyle \pi _{3}={\frac {3}{2}}\sin({\frac {360}{3}})}
π 3 = 1.299038105676657970145584756129404275207103940357785471041855234588949762681600027810859640067936431756719606061555027272477458821249214359108741634070780236396069923926729186550981115075813506192448761484257450177492164988004191001691378478420285348745275896191014287489903034894542100737760629932038067463761097204575710701640392743812359372142969874748342368381216535847874688010418122988440891548250968547863765249583132380335670783737076566229143676252517640533358216521993217274347815376373554674619972363212265589806452991554023153186737988373546994737255879027157790435027172484077780448014436830076173399466281730856066512557064076779745703123294923844606990748216673347110878468792671641139677682025389606048286479258495794767292154648948115366998523815015896613116458830885964900880276393737512516970811610634553457857999822195125972183078771421332595361479060232409486838841413231534260410498721433971667217832916432192792394148303696921223601631524640287512203987208937787304127....... {\displaystyle \pi _{3}=1.299038105676657970145584756129404275207103940357785471041855234588949762681600027810859640067936431756719606061555027272477458821249214359108741634070780236396069923926729186550981115075813506192448761484257450177492164988004191001691378478420285348745275896191014287489903034894542100737760629932038067463761097204575710701640392743812359372142969874748342368381216535847874688010418122988440891548250968547863765249583132380335670783737076566229143676252517640533358216521993217274347815376373554674619972363212265589806452991554023153186737988373546994737255879027157790435027172484077780448014436830076173399466281730856066512557064076779745703123294923844606990748216673347110878468792671641139677682025389606048286479258495794767292154648948115366998523815015896613116458830885964900880276393737512516970811610634553457857999822195125972183078771421332595361479060232409486838841413231534260410498721433971667217832916432192792394148303696921223601631524640287512203987208937787304127.......}

Area of an equilateral triangle will be

R 2 π 3 {\displaystyle R^{2}\pi _{3}}

R -radius of the circle described about an equilateral triangle

The scope of the triangle will be
C = 2 R π 3 cos ⁡ α 2 : α = 360 3 {\displaystyle C={\frac {2R\pi _{3}}{\cos {\frac {\alpha }{2}}}}:\alpha ={\frac {360}{3}}} 
See picture
n=4 it is a square
π 4 = 2 {\displaystyle \pi _{4}=2}

Area of squares will be

R 2 π 4 {\displaystyle R^{2}\pi _{4}}

R -radius of the circle described around the square

Perimeter of the square will be
C = 2 R π 4 cos ⁡ α 2 : α = 360 4 {\displaystyle C={\frac {2R\pi _{4}}{\cos {\frac {\alpha }{2}}}}:\alpha ={\frac {360}{4}}}

See picture

so on pentagon, hexagon .......

π 5 = 2.37764129073788393029109833344845535851424658531437555611826411107538292521298375429698202742702845418974929387850751052246766548442424981578803067578295594696444359492600258179554275230250379513151001423802924795917183008219605036301495148507880140322781045987090044531881885084349467130129921084258543849815492459549445122232219143670592470372285079329288667825627445956811065730153458201304781755172126659167760259062780734991737248477967564240583159502841310815397363785669533897013826055892072659996199853464482719029151537061472938008846993548889178745205678459542206169347872570742004980802724766701370391337077603044283323962631433827549038847681729685293628396669902392009519093871640782190843048473742646406880592139994355352920382189607783603508041226848456648980006775286567800517085352418736253370309354164490169821718538097751802464867296154739841155499816959251406543633373361369339814369337895730123108013172623915093695955211484250762466097770012856809360256952109203831071625 {\displaystyle \pi _{5}=2.37764129073788393029109833344845535851424658531437555611826411107538292521298375429698202742702845418974929387850751052246766548442424981578803067578295594696444359492600258179554275230250379513151001423802924795917183008219605036301495148507880140322781045987090044531881885084349467130129921084258543849815492459549445122232219143670592470372285079329288667825627445956811065730153458201304781755172126659167760259062780734991737248477967564240583159502841310815397363785669533897013826055892072659996199853464482719029151537061472938008846993548889178745205678459542206169347872570742004980802724766701370391337077603044283323962631433827549038847681729685293628396669902392009519093871640782190843048473742646406880592139994355352920382189607783603508041226848456648980006775286567800517085352418736253370309354164490169821718538097751802464867296154739841155499816959251406543633373361369339814369337895730123108013172623915093695955211484250762466097770012856809360256952109203831071625} 
See picture
π 6 = 2.59807621135331594029116951225880855041420788071557094208371046917789952536320005562171928013587286351343921212311005454495491764249842871821748326814156047279213984785345837310196223015162701238489752296851490035498432997600838200338275695684057069749055179238202857497980606978908420147552125986407613492752219440915142140328078548762471874428593974949668473676243307169574937602083624597688178309650193709572753049916626476067134156747415313245828735250503528106671643304398643454869563075274710934923994472642453117961290598310804630637347597674709398947451175805431558087005434496815556089602887366015234679893256346171213302511412815355949140624658984768921398149643334669422175693758534328227935536405077921209657295851699158953458430929789623073399704763003179322623291766177192980176055278747502503394162322126910691571599964439025194436615754284266519072295812046481897367768282646306852082099744286794333443566583286438558478829660739384244720326304928057502440797441787557460825459 {\displaystyle \pi _{6}=2.59807621135331594029116951225880855041420788071557094208371046917789952536320005562171928013587286351343921212311005454495491764249842871821748326814156047279213984785345837310196223015162701238489752296851490035498432997600838200338275695684057069749055179238202857497980606978908420147552125986407613492752219440915142140328078548762471874428593974949668473676243307169574937602083624597688178309650193709572753049916626476067134156747415313245828735250503528106671643304398643454869563075274710934923994472642453117961290598310804630637347597674709398947451175805431558087005434496815556089602887366015234679893256346171213302511412815355949140624658984768921398149643334669422175693758534328227935536405077921209657295851699158953458430929789623073399704763003179322623291766177192980176055278747502503394162322126910691571599964439025194436615754284266519072295812046481897367768282646306852082099744286794333443566583286438558478829660739384244720326304928057502440797441787557460825459}
π 7 = . . . . . . . . . . . {\displaystyle \pi _{7}=...........}
π 8 = . . . . . . . . . . . {\displaystyle \pi _{8}=...........}
. . . . . . . . . . . {\displaystyle ...........}
π 1 2 = 3 {\displaystyle \pi _{1}2=3}
. . . . . . . . . . . {\displaystyle ...........}
π = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837297804995105973173281609631859502445945534690830264252230825334468503526193118817101000313783875288658753320838142061717766914730359825349042875546873115956286388235378759375195778185778053217122680661300192787661119590922 {\displaystyle \pi =3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837297804995105973173281609631859502445945534690830264252230825334468503526193118817101000313783875288658753320838142061717766914730359825349042875546873115956286388235378759375195778185778053217122680661300192787661119590922}
evidence
π = n 2 . s i n ( 360 n ) {\displaystyle \pi ={\frac {n}{2}}.sin({\frac {360}{n}})}
l i m n → ∞ n 2 . s i n ( 360 n ) {\displaystyle lim_{n\to \infty }{\frac {n}{2}}.sin({\frac {360}{n}})}
1 2 l i m n → ∞ n . s i n ( 360 n ) {\displaystyle {\frac {1}{2}}lim_{n\to \infty }n.sin({\frac {360}{n}})}
1 2 l i m n → ∞ s i n ( 360 n ) 1 n {\displaystyle {\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {360}{n}})}{\frac {1}{n}}}}
1 2 l i m n → ∞ s i n ( 360 n ) 1 n . 360 360 {\displaystyle {\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {360}{n}})}{\frac {1}{n}}}.{\frac {360}{360}}}
360 2 l i m n → ∞ s i n ( 360 n ) 360 n {\displaystyle {\frac {360}{2}}lim_{n\to \infty }{\frac {sin({\frac {360}{n}})}{\frac {360}{n}}}}
360 n = t {\displaystyle {\frac {360}{n}}=t}
t →   0 {\displaystyle t\rightarrow \ 0}
180 lim t → 0 s i n t t = 180 0 {\displaystyle 180\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=180^{0}}
180 0 = π ( r a d ) {\displaystyle 180^{0}=\pi (rad)}
-------------------------------------------------------

The following is another new form of PI, which I found in 2004

π = 180. m . s i n ( 1 m ) {\displaystyle \pi =180.m.sin({\frac {1}{m}})}
lim m → ∞ 180. m . s i n ( 1 m ) {\displaystyle \lim _{m\to \infty }180.m.sin({\frac {1}{m}})}
180 lim m → ∞ m . s i n ( 1 m ) {\displaystyle 180\lim _{m\to \infty }m.sin({\frac {1}{m}})}
180 lim m → ∞ s i n ( 1 m ) 1 m {\displaystyle 180\lim _{m\to \infty }{\frac {sin({\frac {1}{m}})}{\frac {1}{m}}}}
1 m = t {\displaystyle {\frac {1}{m}}=t}
m → ∞ {\displaystyle m\to \infty }
t →   0 {\displaystyle t\rightarrow \ 0}
180 lim t → 0 s i n t t = 180 0 {\displaystyle 180\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=180^{0}}
180 0 = π ( r a d ) {\displaystyle 180^{0}=\pi (rad)}
-------------------------------------------------------
π 2 {\displaystyle {\frac {\pi }{2}}}
π 2 = n 2 s i n ( 180 n ) {\displaystyle {\frac {\pi }{2}}={\frac {n}{2}}sin({\frac {180}{n}})}
evidence
l i m n → ∞ n 2 . s i n ( 180 n ) {\displaystyle lim_{n\to \infty }{\frac {n}{2}}.sin({\frac {180}{n}})}
1 2 l i m n → ∞ n . s i n ( 180 n ) {\displaystyle {\frac {1}{2}}lim_{n\to \infty }n.sin({\frac {180}{n}})}
1 2 l i m n → ∞ s i n ( 180 n ) 1 n {\displaystyle {\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {180}{n}})}{\frac {1}{n}}}}
1 2 l i m n → ∞ s i n ( 180 n ) 1 n . 180 180 {\displaystyle {\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {180}{n}})}{\frac {1}{n}}}.{\frac {180}{180}}}
90 lim m → ∞ s i n ( 1 m ) 1 m {\displaystyle 90\lim _{m\to \infty }{\frac {sin({\frac {1}{m}})}{\frac {1}{m}}}}
180 n = t {\displaystyle {\frac {180}{n}}=t}
n → ∞ {\displaystyle n\to \infty }
t →   0 {\displaystyle t\rightarrow \ 0}
90 lim t → 0 s i n t t = 90 0 {\displaystyle 90\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=90^{0}}
90 0 = π 2 ( r a d ) {\displaystyle 90^{0}={\frac {\pi }{2}}(rad)}
-------------------------------------------------------
π 3 {\displaystyle {\frac {\pi }{3}}}
π 3 = n 2 s i n ( 120 n ) {\displaystyle {\frac {\pi }{3}}={\frac {n}{2}}sin({\frac {120}{n}})}
evidence
l i m n → ∞ n 2 . s i n ( 120 n ) {\displaystyle lim_{n\to \infty }{\frac {n}{2}}.sin({\frac {120}{n}})}
1 2 l i m n → ∞ n . s i n ( 120 n ) {\displaystyle {\frac {1}{2}}lim_{n\to \infty }n.sin({\frac {120}{n}})}
1 2 l i m n → ∞ s i n ( 120 n ) 1 n {\displaystyle {\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {120}{n}})}{\frac {1}{n}}}}
1 2 l i m n → ∞ s i n ( 120 n ) 1 n . 120 120 {\displaystyle {\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {120}{n}})}{\frac {1}{n}}}.{\frac {120}{120}}}
60 lim n → ∞ s i n ( 120 n ) 120 n {\displaystyle 60\lim _{n\to \infty }{\frac {sin({\frac {120}{n}})}{\frac {120}{n}}}}
120 n = t {\displaystyle {\frac {120}{n}}=t}
n → ∞ {\displaystyle n\to \infty }
t →   0 {\displaystyle t\rightarrow \ 0}
60 lim t → 0 s i n t t = 60 0 {\displaystyle 60\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=60^{0}}
60 0 = π 3 ( r a d ) {\displaystyle 60^{0}={\frac {\pi }{3}}(rad)}
-------------------------------------------------------
π 4 {\displaystyle {\frac {\pi }{4}}}
π 4 = n 2 . s i n ( 90 n ) {\displaystyle {\frac {\pi }{4}}={\frac {n}{2}}.sin({\frac {90}{n}})}
evidence
l i m n → ∞ n 2 . s i n ( 90 n ) {\displaystyle lim_{n\to \infty }{\frac {n}{2}}.sin({\frac {90}{n}})}
1 2 l i m n → ∞ n . s i n ( 90 n ) {\displaystyle {\frac {1}{2}}lim_{n\to \infty }n.sin({\frac {90}{n}})}
1 2 l i m n → ∞ s i n ( 90 n ) 1 n {\displaystyle {\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {90}{n}})}{\frac {1}{n}}}}
1 2 l i m n → ∞ s i n ( 90 n ) 1 n . 90 90 {\displaystyle {\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {90}{n}})}{\frac {1}{n}}}.{\frac {90}{90}}}
90 2 l i m n → ∞ s i n ( 90 n ) 90 n {\displaystyle {\frac {90}{2}}lim_{n\to \infty }{\frac {sin({\frac {90}{n}})}{\frac {90}{n}}}}
90 n = t {\displaystyle {\frac {90}{n}}=t}
n → ∞ {\displaystyle n\to \infty }
t →   0 {\displaystyle t\rightarrow \ 0}
45 lim t → 0 s i n t t = 45 0 {\displaystyle 45\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=45^{0}}


45 0 = π 4 ( r a d ) {\displaystyle 45^{0}={\frac {\pi }{4}}(rad)}
-------------------------------------------------------
π 5 {\displaystyle {\frac {\pi }{5}}}
π 5 = n 2 s i n ( 72 n ) {\displaystyle {\frac {\pi }{5}}={\frac {n}{2}}sin({\frac {72}{n}})}
evidence
l i m n → ∞ n 2 . s i n ( 72 n ) {\displaystyle lim_{n\to \infty }{\frac {n}{2}}.sin({\frac {72}{n}})}
1 2 l i m n → ∞ n . s i n ( 72 n ) {\displaystyle {\frac {1}{2}}lim_{n\to \infty }n.sin({\frac {72}{n}})}
1 2 l i m n → ∞ s i n ( 72 n ) 1 n {\displaystyle {\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {72}{n}})}{\frac {1}{n}}}}
1 2 l i m n → ∞ s i n ( 72 n ) 1 n . 72 72 {\displaystyle {\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {72}{n}})}{\frac {1}{n}}}.{\frac {72}{72}}}
72 2 l i m n → ∞ s i n ( 72 n ) 72 n {\displaystyle {\frac {72}{2}}lim_{n\to \infty }{\frac {sin({\frac {72}{n}})}{\frac {72}{n}}}}
72 n = t {\displaystyle {\frac {72}{n}}=t}
n → ∞ {\displaystyle n\to \infty }
t →   0 {\displaystyle t\rightarrow \ 0}
72 2 l i m t → 0 s i n t t = 72 2 {\displaystyle {\frac {72}{2}}lim_{t\to \mathbf {0} }{\frac {sint}{t}}={\frac {72}{2}}}
72 2 = π 5 ( r a d ) {\displaystyle {\frac {72}{2}}={\frac {\pi }{5}}(rad)}
-------------------------------------------------------
π 8 {\displaystyle {\frac {\pi }{8}}}
π 8 = n 2 s i n ( 45 n ) {\displaystyle {\frac {\pi }{8}}={\frac {n}{2}}sin({\frac {45}{n}})}
-------------------------------------------------------

The following is another new form of PI, which I found in 2004

π = n . sin ⁡ ( 180 n ) {\displaystyle \pi =n.\sin({\frac {180}{n}})}
evidence
  l i m n → ∞ n . s i n ( 180 n ) {\displaystyle \ lim_{n\to \infty }n.sin({\frac {180}{n}})}
  l i m n → ∞ s i n ( 180 n ) 1 n {\displaystyle \ lim_{n\to \infty }{\frac {sin({\frac {180}{n}})}{\frac {1}{n}}}}
  l i m n → ∞ s i n ( 180 n ) 1 n . 180 180 {\displaystyle \ lim_{n\to \infty }{\frac {sin({\frac {180}{n}})}{\frac {1}{n}}}.{\frac {180}{180}}}
180 lim m → ∞ s i n ( 180 n ) 180 n {\displaystyle 180\lim _{m\to \infty }{\frac {sin({\frac {180}{n}})}{\frac {180}{n}}}}
180 n = t {\displaystyle {\frac {180}{n}}=t}
n → ∞ {\displaystyle n\to \infty }
t →   0 {\displaystyle t\rightarrow \ 0}
180 lim t → 0 s i n t t = 180 0 {\displaystyle 180\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=180^{0}}
180 0 = π ( r a d ) {\displaystyle 180^{0}=\pi (rad)}
-------------------------------------------------------
π 2 {\displaystyle {\frac {\pi }{2}}}
π 2 = n . sin ⁡ ( 90 n ) {\displaystyle {\frac {\pi }{2}}=n.\sin({\frac {90}{n}})}
-------------------------------------------------------
π 3 {\displaystyle {\frac {\pi }{3}}}
π 3 = n . sin ⁡ ( 60 n ) {\displaystyle {\frac {\pi }{3}}=n.\sin({\frac {60}{n}})}
-------------------------------------------------------
π 5 {\displaystyle {\frac {\pi }{5}}}
π 5 = n . sin ⁡ ( 36 n ) {\displaystyle {\frac {\pi }{5}}=n.\sin({\frac {36}{n}})}
-------------------------------------------------------

The following is another new form of PI, which I found in 2004

π = 360. m . s i n ( 1 2 m ) {\displaystyle \pi =360.m.sin({\frac {1}{2m}})}
lim m → ∞ 360. m . s i n ( 1 2 m ) {\displaystyle \lim _{m\to \infty }360.m.sin({\frac {1}{2m}})}
360 lim m → ∞ m . s i n ( 1 2 m ) {\displaystyle 360\lim _{m\to \infty }m.sin({\frac {1}{2m}})}
360 lim m → ∞ s i n ( 1 2 m ) 1 m {\displaystyle 360\lim _{m\to \infty }{\frac {sin({\frac {1}{2m}})}{\frac {1}{m}}}}
360 lim m → ∞ s i n ( 1 2 m ) 1 m 1 2 1 2 {\displaystyle 360\lim _{m\to \infty }{\frac {sin({\frac {1}{2m}})}{\frac {1}{m}}}{\frac {\frac {1}{2}}{\frac {1}{2}}}}
360. 1 2 lim m → ∞ s i n ( 1 2 m ) 1 2 m {\displaystyle 360.{\frac {1}{2}}\lim _{m\to \infty }{\frac {sin({\frac {1}{2m}})}{\frac {1}{2m}}}}
1 2 m = t {\displaystyle {\frac {1}{2m}}=t}
m → ∞ {\displaystyle m\to \infty }
t →   0 {\displaystyle t\rightarrow \ 0}
180 lim t → 0 s i n t t = 180 0 {\displaystyle 180\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=180^{0}}
180 0 = π ( r a d ) {\displaystyle 180^{0}=\pi (rad)}

For the numerical values ​​of PI recommend free extra precision calculator Harry-J-Smith XP,XM,….. recommend m=1.0E+10000000 to the success of the calculator you need to install netframework2.0



(Рајко Велимировић (talk) 10:02, 21 December 2011 (UTC))

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