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What is exactly meant by "angle subtended by all sides of a polygon at its center" ? I encountered these while solving this problem:

How can we determine the number of sides of a polygon,given the angles subtended by all the side at its center are equal to $72 ^\circ$?

Could somebody please explain (with a figure) what is meant to calculate in this question?

EDIT: I tried to Google some help, which took me to this page.I think this is what meant in this question.If so why all sides is mention in this question? and why it is supposed that the polygon will have equal sides?

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up vote 2 down vote accepted

Here's a rough diagram showing the center of the polygon and some sides and indicating the angle that is subtended by a side.


You are given that the measure of all such angles in a particular polygon is 72° and asked to determine the number of sides in the polygon.

edit: As to why "all sides" is used in the problem, I'd guess it's a translation/language issue and that "each side" is a more appropriate description of what was intended.

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I really appreciate your efforts but I guess i have edited the question while you are writing the answer,so I guess it's the same idea explained in that link.But I am not sure why all sides meant and why it is supposed that all sides will have equal lengths ?! – Quixotic Jan 5 '11 at 11:45
@Debanjan: After seeing your edit, I added a note at the end of my answer regarding the "all sides" language--I think it probably should have said "each side." I don't see where you're getting the equal lengths part, though. – Isaac Jan 5 '11 at 11:47
Thanks a lot, I am not sure if I am correct or not but what about this polygon: or we cannot have a center for this ? – Quixotic Jan 5 '11 at 11:49
@Debanjan: I think, for non-regular polygons, "center" is probably not well-defined. (There are thousands of known centers for triangles.) "centroid" (center of mass assuming uniform distribution of mass across the area of the polygon) might be a reasonable choice for a "center," at least for convex polygons. – Isaac Jan 5 '11 at 11:53
Hm so we have to assume regular polygons only. Thanks Issac :) – Quixotic Jan 5 '11 at 11:55

If there are 360 degrees in a circle...

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It seems like I haven't understood the problem actually from your answer it seems like you are hinting for $\frac{360}{72}$ but the 3D/2D diagram for this problem is not clear in my head,may be due the words I am confusing. – Quixotic Jan 5 '11 at 11:19
That was exactly my point. I didn't seen anything about 3D in the question. The reason the page you link to calls for equal sides is so you can divide by n. If the sides are not equal (and even if they are equal if the angles are not) they may not all subtend the same angle. – Ross Millikan Jan 5 '11 at 15:42

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