數學知識交流---七角星內角和

∠B + ∠D + ∠G + ∠ION = 360 (∠ sum of polygon)
∠AON = 360 - (∠B + ∠D + ∠G)
∠C + ∠E + ∠H + ∠ONM = 360 (∠ sum of polygon)
∠ONM = 360 - (∠C + ∠E + ∠H)
∠B + ∠D + ∠F + ∠NMI = 360 (∠ sum of polygon)
∠NMI = 360 - (∠B + ∠D + ∠F)
∠C + ∠E + ∠G + ∠MIK = 360 (∠ sum of polygon)
∠MIK = 360 - (∠C + ∠E + ∠G)
∠D + ∠F + ∠H + ∠IKJ = 360 (∠ sum of polygon)
∠IKJ = 360 - (∠D + ∠F + ∠H)
∠B + ∠E + ∠G + ∠KJI = 360 (∠ sum of polygon)
∠KJI = 360 - (∠B + ∠E + ∠G)
∠C + ∠F + ∠H + ∠JIO = 360 (∠ sum of polygon)
∠JIO = 360 - (∠C + ∠F + ∠H)

Consider the polygon MNOIJKL.
By ∠ sum of polygon,
∠AON + ∠ONH + ∠NMI + ∠MIK + ∠IKJ + ∠KJI + ∠JIO = (7 - 2)*180
360 - (∠B + ∠D + ∠G) + 360 - (∠C + ∠E + ∠H) + 360 - (∠B + ∠D + ∠F) + 360 - (∠C + ∠E + ∠G) + 360 - (∠D + ∠F + ∠H) + 360 - (∠B + ∠E + ∠G) + 360 - (∠C + ∠F + ∠H) = 900
3(∠B + ∠C + ∠D + ∠E + ∠F + ∠G + ∠H) = 1620
∠B + ∠C + ∠D + ∠E + ∠F + ∠G + ∠H = 540

In this graph:

http://s1099.photobucket.com/albums/g395/jasoncube/?action=view¤t=e284b6b6.png


圖片參考：http://imgcld.yimg.com/8/n/HA00016323/o/701110070076413873473130.jpg

when B,C,D,E are infinitively close to each others

and F,G,H are infinitively close to each others.

∠B=∠C=∠D=∠E=∠F=∠H=90° and ∠G=0°.

So, the sum of angles of 7-angles star is,

=90°×6+0°×1

=540°