What do the measures of the interior angles approach as the number of sides increases?

Answer 1

As the number of sides increases, the number of interior angles increases,

and if the polygon is regular, the measure of each individual interior angle

also increases, always getting closer and closer to 180 degrees each.

Eventually, if this keeps going to a ridiculous extreme, the figure keeps getting

closer to the special polygon with an infinite number of sides and an infinite

number of 180-degree interior angles, called a "circle".

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The formula for calculating the interior angle of one angle of a regular polygon is 180*(n-2)/n, where n is the number of sides. If you do a range of calculations for polygons with increasing numbers of sides, (you could stop at 1,000!) you will observe the truth of the explanatory statement given in the first paragraph above.