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Our teacher said this was a tricky question but I do not understand why. If I have a circle that has ten dots on it and I have to make as many polygons using those dots, the answer should be 10. Connecting each dot starting with triangle, to square, all the way to the max of a 10-sided polygon should give me only 10 polygons that I can make. Am I missing something here?

user510
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  • There are only eight numbers from three through ten. Where are you getting ten? There are ten nine sided polygons, one missing each vertex, assuming you prohibit polygons which have crossing sides. – Ross Millikan Nov 20 '16 at 05:58
  • Are you considering polygons different if they have different lengths/angles and the same number of sides? If so are the dots equally spaced? And if so do you count rotations of the same shape as different? – Ian Miller Nov 20 '16 at 06:15
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    Do you mean sphere as in your title or circle as in the body? – Ross Millikan Nov 20 '16 at 06:43
  • i mean a circle that has ten dots on it and you have to connect the dots and come up with as many polygons. Then , you have to give number of regular polygons, and the number of irregular polygons that can be formed – user510 Nov 20 '16 at 15:02

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There are several ways to read the question which lead to different answers. One way is to ask how many disjoint polygons you can form simultaneously. This would be the number of triangles in a triangulation of the decagon. Another would be to ask how many different polygons that do not have intersecting sides you can form. In this case there is one for any combination of at least three vertices. Another would be to ask how many different polygons you can form if you allow the sides to intersect. In this case for each set of four vertices there are three, not one, polygon that can be formed. For $n$ vertices you can make $\frac {(n-1)!}2$ polygons this way.

Ross Millikan
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