By inspection from smaller polygons,
4 sides: 2S+D , where S is a side, and D is the diameter
6 sides 2S+2S1+D, where S1 is the line of intermediate length
8 sides 2S+2S1+2S2+D, etc.
Since all the vertices are on the circumference, these can all be seen as right triangles, with D as the hypotenuse, so their measure is:
2^2+2*(2*sin(12))^2+2*(2*sin(24))^2+2*(2*sin(36))^2+2*(2*sin(48))^2+2*(2*sin(60))^2+2*(2*sin(72))^2+2*(2*sin(84))^2 = 34, plus
2*(2*sin(6))^2+2*(2*sin(18))^2+2*(2*sin(30))^2+2*(2*sin(42))^2+2*(2*sin(54))^2+2*(2*sin(66))^2+2*(2*sin(78))^2 =26
for a total of 60.

Posted by broll
on 20180417 13:36:26 