I'm talking about the short version of the area of a pentagram inscribed in a circle when the radius is given and so Area $=1.123r^2$.
I only got this formula from a book but I researched about it and nothing came up.
I tried to apply the short formula in other problems online and I still got the correct answer.
$\endgroup$ 61 Answer
$\begingroup$A regular star with five vertices ($\bigstar$) and unit circumradius can be seen as the disjoint union of ten triangles having a side with unit length and the angles adjacent to it equal to $\frac{\pi}{5}$ and $\frac{\pi}{10}$. By the sine theorem, the area of a triangle made that way is: $$\frac{1}{2}\cdot\frac{\sin\frac{\pi}{10}\sin\frac{\pi}{5}}{\sin\frac{3\pi}{10}}=\frac{1}{2}\cdot\frac{1}{\cot\frac{\pi}{5}+\cot\frac{\pi}{10}}$$ hence the area of the "pentagram" inscribed in a circle with radius $R$ is given by: $$ \frac{5R^2}{\cot\frac{\pi}{5}+\cot\frac{\pi}{10}}=\frac{5R^2}{2}\sqrt{\frac{25-11\sqrt{5}}{2}} \approx \color{red}{1.12257}\,R^2.$$
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