By what factor do you need to multiply the equation of a circumscribed polygon so that it shrinks into an inscribed polygon.

Question

The factor that is needed to be multiplied can be easily figured out for any given regular polygon, but is there an equation that takes the number of sides and produces a number that when multiplied to the equation of a circumscribed polygon, the polygon will shrink and become inscribed.

Answer 1

That's an interesting question. I understand that its asking for the scale factor that shrinks a circumscribed regular polygon with N sides (preimage) to an inscribed regular polygon with N sides (image). The scale factor will be the ratio of the inscribed polygon’s radius to the circumscribed polygon’s radius. An inscribed polygon has the same radius as the circle. But for a circumscribed regular polygon, the radius of the circle equals the apothem. To see this, I drew the image and preimage for a square (N=4). Note that each central angle would be 90 degrees, which was found using 360/4. For an N-sided regular polygon, the central angle would be 360/N.

Looking at just part of that figure, I drew the apothem to form a right triangle. The apothem bisects the central angle which reduces its measure to 360/(2N) or just 180/N. Therefore,

Scale Factor = (radius of the inscribed regular polygon) / (radius of the circumscribed regular polygon)

Note again that the radius of the circle equals the radius of the inscribed regular polygon which equals the apothem of the circumscribed regular polygon (I wish that I could include my drawing here like I would in a lesson).

Scale Factor = apothem / hypotenuse

Scale Factor = cos(180/N)

I hope this is useful to you, and please let me know if you have any other questions.