Decagon

The constructible regular 10-sided Polygon with Schläfli Symbol $\{10\}$. The Inradius $r$, Circumradius $R$, and Area can be computed directly from the formulas for a general regular Polygon with side length $s$ and $n=10$ sides,

$$r = {\textstyle{1\over 2}}s\cot\left({\pi\over 10}\right)={\textstyle{1\over 2}}\sqrt{25-10\sqrt{5}}\,s$$ (1)

$$R = {\textstyle{1\over 2}}s\csc\left({\pi\over 10}\right)={\textstyle{1\over 2}}(1+\sqrt{5})s=\phi s$$ (2)

$$A = {\textstyle{1\over 4}}n s^2\cot\left({\pi\over 10}\right)={\textstyle{5\over 2}}\sqrt{5+2\sqrt{5}}\,s^2.$$ (3)

Here, $\phi$ is the Golden Mean.

See also Decagram, Dodecagon, Trigonometry Values Pi/10, Undecagon

References

Dixon, R. Mathographics. New York: Dover, p. 18, 1991.