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
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, TheSimpliFire, YuiTo Cheng, MathOverview, Shailesh
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.