Is it possible to cover all area of a circle of radius r>0 with infinite lines starting from the center?


closed as off-topic by José Carlos Santos, TheSimpliFire, YuiTo Cheng, MathOverview, Shailesh Jun 11 at 14:10

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    $\begingroup$ Do you mean 'from center'? $\endgroup$ – ajotatxe Jun 11 at 12:58
  • $\begingroup$ I don't know what "from starting from radius" means. Does "infinite lines" mean "infinitely many lines"? $\endgroup$ – Gerry Myerson Jun 11 at 13:03
  • $\begingroup$ Do yo mean infinite lines or infinitely many lines ? $\endgroup$ – Yves Daoust Jun 11 at 13:09
  • $\begingroup$ Do you mean "arcs" instead of lines? Then it's 2 times the integral of the half-circle $\endgroup$ – Klangen Jun 11 at 13:24
  • $\begingroup$ yes, i mean from the center $\endgroup$ – Horacio Passardi Jun 11 at 14:00

Yes, it is (assuming you mean "start from the center", and you mean "infinitely many lines"). For each point on the circle, take the line going through the origin and that point. Any point in the disc lies on one of these lines, so their union covers the entire disc (and in fact the entire plane).


Every point in the disk is on some radius, so the union of all the radii covers the disk.

There are infinitely many radii, and all are needed to complete the cover.

But note -- the set of radii is an uncountable set. In other words, you can't list them as a sequence.


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