Polygons with convergent area

Question

Suppose that there is an equilateral triangle of area 10, call this $A_0$, we then trisect each side of this triangle, then cut off the corners, giving the polygon $A_1$, what is the area of this polygon, we then do this again to get $A_2$ (trisect sides, and cut corners), what is the area of this polygon? What is the area of $A_\infty$.

With this problem, I initially tried to generalise side length for $A_n$, then used the area of a polygon formula to find the area, I was wondering if anyone could come up with any other ideas, or alternatively do it the same way I did, to compare answers. Best of luck!

You have continuously asked this very question HERE, I believe 3 times in a row---->https://math.stackexchange.com/questions/2665051/convergent-area?noredirect=1#comment5504953_2665051 — , Feb 24 '18 at 22:46

score 0 · Answer 1 · answered Feb 24 '18 at 22:59

0

The answer for $A_n$ as n goes to infinity is $0$

Note that $$A_0 = 10$$

$$A_1 = 10(2/3)$$ $$A_2= 10(2/3)^2$$ $$A_3= 10(2/3)^3$$ $$.$$ $$.$$ $$.$$ $$A_n= 10(2/3)^{n}$$ $$ A_{\infty } =lim _{n\to \infty} 10(2/3)^n = 0 $$

answered Feb 24 '18 at 22:59

Mohammad Riazi-Kermani

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So you are not cutting away all the corners of the new triangles each time? My answer was based on my assumption that you do the cutting on remaining little triangles as you did with the original one. – Mohammad Riazi-Kermani Feb 25 '18 at 18:53
Yes that was the idea – seasons23 Mar 04 '18 at 11:34

Polygons with convergent area

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