==========================================================================================================
triangle
x = {+1}

x - 1 = 0

https://www.wolframalpha.com/input?i2d=true&i=x+-+1+%3D+0
==========================================================================================================

pentagon
x = {+1.618, -0.618}

{(5 - 2) x 180} / 5 = 540 / 5 = 108 / (5 - 2) = 108 / 3 = 36 / 2 = 18
1 / [ {sin(18) = 0.30901699437494742410229341718282 x 2} = 0.61803398874989484820458683436564 ] = 1.61803398874989484820458683436564
1 / [ {sin(54) = 0.80901699437494742410229341718282 x 2} = 1.61803398874989484820458683436564 ] = 0.61803398874989484820458683436564

{1 - sqrt(5)} / 2
{1 + sqrt(5)} / 2

x2 - x - 1 = 0

https://www.wolframalpha.com/input?i2d=true&i=x2+-+x+-+1+%3D+0

3 triangles x 180 @ = 540 tot angles / 5 = 108 degrees apex at center
180 - 108 = 72
72 / 2 = 36

cos(18) = a/h = a/1 = a = 0.95105651629515357211643933337938
2 x 0.95105651629515357211643933337938 = 1.9021130325903071442328786667588

1.9021130325903071442328786667588^2 = 3.6180339887498948482045868343656
3.6180339887498948482045868343656 - 2 = 1.6180339887498948482045868343656

cos(54) = a/h = a/1 = a = 0.58778525229247312916870595463907
2 x 0.58778525229247312916870595463907 = 1.1755705045849462583374119092781

1.1755705045849462583374119092781^2 = 1.3819660112501051517954131656344
1.3819660112501051517954131656344 - 1 = 0.3819660112501051517954131656344
sqrt(0.3819660112501051517954131656344) = 0.6180339887498948482045868343656

==========================================================================================================

septagon
x = {-0.80194, +0.55496, +2.2470}

X3 - 2X2 - X1 + 1 = 0

https://www.wolframalpha.com/input?i2d=true&i=x3+%E2%80%93+2x2+-+x+%2B+1+%3D+0
==========================================================================================================

nonagon
x = {-1, -0.53209, +0.65270, +2.8794}
<[{(<9 - 2> x 180) / 9} / <9 - 2>] / 2>
2.8794 = sin{(7 x 180) / 9} = sin{(7 x 20) / 7} = 1 / [{sin(20 / 2) x 2}] = 1 / {sin(10) x 2}
----------------------------------------------------------------------------------
-1
1 - 1/2 (1 - i sqrt(3)) (1/2 (1 + i sqrt(3)))^(1/3) - (1/2 (1 + i sqrt(3)))^(2/3)
1 - (1 - i sqrt(3))/(2^(2/3) (1 + i sqrt(3))^(1/3)) - (1 + i sqrt(3))^(4/3)/(2 2^(1/3))
1 + 1/(1/2 (1 + i sqrt(3)))^(1/3) + (1/2 (1 + i sqrt(3)))^(1/3)
----------------------------------------------------------------------------------------

x4 - 2x3 – 3x2 + x + 1 = 0

https://www.wolframalpha.com/input?i2d=true&i=x4+-+2x3+%E2%80%93+3x2+%2B+x+%2B+1+%3D+0
==========================================================================================================

11-gon
x = {-1.20362, -0.594351, +0.521109, +0.763521, +3.51334}

x5 - 3x4 - 3x3 + 4x2 + x - 1 = 0

https://www.wolframalpha.com/input?i2d=true&i=x5+-+3x4+-+3x3+%2B+4x2+%2B+x+-+1+%3D+0
==========================================================================================================

13-gon
x = {-1.41002, -0.667993, -0.514964, +0.564681, +0.880181, +4.14811}

x6 - 3x5 - 6x4 + 4x3 + 5x2 - x - 1 = 0

https://www.wolframalpha.com/input?i2d=true&i=x6+-+3x5+-+6x4+%2B+4x3+%2B+5x2+-+x+-+1+%3D+0
==========================================================================================================

15-gon
------------------------------------------------
<[{(<15 - 2> x 180) / 15} / <15 - 2>] / 2>
<[{(<13> x 180) / 15} / <13>] / 2>
<[{2340 / 15} / 13] / 2>
<[156 / 13] / 2>
<[12] / 2>
6
1 / {sin(6) x 2} = +4.7834
-------------------------------------------------
+1
-1/2 ± sqrt(5)/2
1/2 × (2 ± sqrt(5) - sqrt{ 15 ± 6 × sqrt(5) })
1/2 × (2 ± sqrt(5) + sqrt{ 15 ± 6 × sqrt(5) })
--------------------------------------------------
the ± is tied together so that all of them switch polarity at the same time
----------------------------------------
+1
-1.6180
+0.61803
-0.74724
+0.51117
-0.54732
+4.7834
-------------------------------------------------

x7 - 4x6 - 6x5 + 10x4 + 5x3 - 6x2 - x + 1 = 0

https://www.wolframalpha.com/input?i2d=true&i=x7+-+4x6+-+6x5+%2B+10x4+%2B+5x3+-+6x2+-+x+%2B+1+%3D+0
==========================================================================================================

17-gon
x8 - 4x7 - 10x6 + 10x5 + 15x4 - 6x3 - 7x2 + x + 1 = 0
------------------------------------------------
<[{(<17 - 2> x 180) / 17} / <17 - 2>] / 2>
<[{(<15> x 180) / 17} / <15>] / 2>
<[{2700 / 17} / 15] / 2>
<[158.82353 / 15] / 2>
<[10.59] / 2>
5.2941176470588235294117647058824
1 / {sin(5.2941176470588235294117647058824) x 2} = +5.4189757237297103662675177164564

5.4189757237297103663

https://www.wolframalpha.com/input?i2d=true&i=x8+-+4x7+-+10x6+%2B+10x5+%2B+15x4+-+6x3+-+7x2+%2B+x+%2B+1+%3D+0
==========================================================================================================
==========================================================================================================
https://www.wolframalpha.com/input?i=%281%2F4%29sqrt%7B34+-+sqrt%2817%29+-+sqrt%5B34+-+2sqrt%2817%29%5D+-+2sqrt%2817+%2B+3sqrt%2817%29+%2B+sqrt%5B170+-+26sqrt%2817%29%5D+-+4sqrt%5B34+%2B+2sqrt%2817%29%5D%29%7D

(1/4)sqrt{34 - sqrt(17) - sqrt[34 - 2sqrt(17)] - 2sqrt(17 + 3sqrt(17) + sqrt[170 - 26sqrt(17)] - 4sqrt[34 + 2sqrt(17)])} =
= 1.0630276433826376254829304357719422346838295622886161900761013660


https://web.archive.org/web/20150909225801/http://www.uwgb.edu/dutchs/PSEUDOSC/17-gon.HTM

==========================================================================================================

3x60=180

5 x 36 = 180

3x5=15 x 12 = 180

5x9=45 x 4 = 180

45-gon
------------------------------------------------
<[{(<45 - 2> x 180) / 45} / <45 - 2>] / 2>
<[{(<43> x 180) / 45} / <43>] / 2>
<[{7740 / 45} / 43] / 2>
<[172 / 43] / 2>
<[4] / 2>
2
1 / {sin(2) x 2} = +14.326854173921910568260819108083