What is the relation between inradius and circumradius of a hexagon

Question

Let R and r be respectively circumradius and inradius of a hexagon, I would like to know the math relation between R and r. Thanks,

Answer 1

Assuming we are dealing with a regular hexagon, $$\frac{R}{r}=\frac{2}{\sqrt{3}}$$ since such a ratio is just the ratio between the side and the height of an equilateral triangle.

Answer 2

The circumscribed radius $R$ & the inscribed radius $r$ of any regular n-gon are generally co-related as follows$$\bbox [4pt, border: 1px solid blue;] {R=r\sec\frac{\pi}{n}}$$ As per your question, for regular hexagon substitute $n=6$ in the above generalized relation, we get $$\frac{R}{r}=\sec\frac{\pi}{6}=\frac{2}{\sqrt3}$$