A simple fractal. How to find the area of it? (only the arms of the star)
Working with pentagrams is quite complicated, I can not solve this.
A simple fractal. How to find the area of it? (only the arms of the star)
Working with pentagrams is quite complicated, I can not solve this.
The easiest method to calculate only the "arms" of the stars is to subtract the areas of all the larger triangles (for lack of better wording) in each iteration from the area of the entire pentagon.
First of all, we calculate the area of the entire pentagon (assuming the side length is 1) to be $5\times\frac{1}{2}\times\left(\frac{sin{54}}{sin{72}}\right)^2\times\sin{72}$
Since pentagrams are well known for their relationship with the golden ratio ($\phi=\frac{1+\sqrt{5}}{2}$), the side lengths can be calculated easily using these identities.
The total area of the obtuse triangles in the first iteration is $5\times\frac{1}{2}\times(\phi-1)^2\times\sin{108}$
Next, we need to calculate the ratio between the area of each iteration of obtuse triangles. Since the triangles are similar, the ratio r (smaller/larger) can be simplified to $(2-\phi)^2$ Since $r>1$, this is identified as a geometric series and therefore the area of all the obtuse triangles can be found using the formula $\frac{a}{1-r}$
$A_{total}=\frac{a}{1-r}$ $=\frac{5\times\frac{1}{2}\times(\phi-1)^2\times\sin{108}}{1-(2-\phi)^2}$
We then subtract this value from the area of the entire pentagon to get $5\times\frac{1}{2}\times\left(\frac{sin{54}}{sin{72}}\right)^2\times\sin{72}-\frac{5\times\frac{1}{2}\times(\phi-1)^2\times\sin{108}}{1-(2-\phi)^2}$
The total area of all the "arms" of the star $\approx0.65716$
I do not think the last equation can be simplified further, but if anyone finds a way, please leave a comment.