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

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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 $10$.