reply on: [Not-so-daily puzzle] Purple area \ stacker news ~science

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Can any of the lines from one vertex to another (passing through the middle point), be assumed to be straight?

If the line passes through the center point, yes. If the line goes through any other point, it'll have a kink, as seen in the figure.

Edit: let me know if I misunderstood your question.

I guess the question is, are the green lines intersecting chords through one and the same point?

Oh, got it.

Yes. The answer is yes, those are straight lines. I was a bit too jetlagged when answering #808331 or #808835.

I think the area of the magenta region is

. Let me know if that's correct, and I'll explain.

If we look at the upper blue region:

The area is the sum of the components:

Adding the same (with

instead of

) for the lower blue region:

The apothem formula gives

Simple trig says

Those last four equations solve for

, or total area of polygon

As seen in @0xbitcoiner's drawing, the total area is the sum

Therefore,

I'm not going to be the one checking those calculations! 😎👍

Correct, the area of the magenta region equals $10$.