Quandaries and Queries


Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees.  What is the greatest number of sides the polygon could have?


Each polygon has different internal angle sum.  The triangle has 180 degrees, a quadrilateral 360 degrees, pentagon 540 degrees etc.  It's to do with the number of tringles each polygon can be split into.

thank you




Hi Victoria,

I would suggest that this, like many problems, is a lot easier to do using exterior angles.

For the interior angle to be a whole number, the exterior angle is also a whole number. The sum of the exterior angles of a simple (not self-intersecting) polygon is 360 degrees, for all possible numbers of sides.

To keep all the interior angles obtuse, the polygon has to be convex. If all the angles are the smallest possible whole number, they will all be 1 degree. You can finish this off, knowing that all the angles are the same (1 degree) size and that the sum is 360 degrees.

One can check that this makes all interior angles 180 degrees - 1 degree.



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