Question 979233
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If *[tex \Large n] is the number of sides and *[tex \Large n|(n-2)180], then the measure of each interior angle is an integer.


As *[tex \Large n] increases, the measure of the interior angles increases because *[tex \Large \frac{(n\ -\ 2)180}{n}] gets closer to 180 as *[tex \Large n] gets larger.  A circle is the limiting shape as *[tex \Large n] increases without bound so a circle is an infinite-sided polygon with an infinite number of vertices with 180 degree interior angles.  It was this idea that allowed Archimedes to approximate *[tex \Large \pi] by sandwiching a circle between two 96-sided polygons, one inscribed and the other circumscribed.


As *[tex \Large n] increases, the measure of the exterior angles decreases because *[tex \Large \frac{360}{n}] gets smaller as *[tex \Large n] gets larger.


As *[tex \Large n] increases, the total of the measures of the interior angles increases because *[tex \Large (n\ -\ 2)180] gets larger as *[tex \Large n] gets larger.


As *[tex \Large n] increases, the total of the measures of the exterior angles remains constant because *[tex \Large 360] does not change as *[tex \Large n] gets larger.  Another way to put it is, no matter how many increasingly smaller turns you make, you still only go around one circle when you get back to where you started.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it

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