1

I have asked this question on a couple of other mathematics forums without solution so thought I might try it out here. Imagine you have an ellipse of half major axis of $1.5$ and half minor axis of $1$ and wished to inscribe within it an equilateral polygon of $12$ sides. That would make $3$ sides for each quadrant and $4$ vertices touching the perimeter in each quadrant.

What would be the length of one of the sides?

For an ellipse of zero eccentricity the formula $$R\sqrt{2\left[1-\cos\left(\frac{360}{n}\right)\right]}$$ will suffice but what about ellipses of greater eccentricity?

N. F. Taussig
  • 76,571
Steven
  • 59

1 Answers1

1

I chose two points $$\left(x,\sqrt{1-(x/1.5)^2}\right),\left(1.5\sqrt{1-y^2},y\right)$$ as well as $(0,1)$ and $(1.5,0)$. Assume the three distances are equal. You get messy equations from Pythagoras. WolframAlpha can solve them. It says $x$ obeys a degree-8 polynomial equation, whose coefficients are 12-digit numbers.

Empy2
  • 50,853