Probability of a circle landing in a hexagon I recently saw this question in my math textbook:

What that probability a circle of radius $6\sqrt{3}$, when thrown to a random place in an infinite sheet of regular hexagons of side length $36$, fits exactly inside the hexagon?

My first question was, are you throwing the circle? (Maybe not so helpful)
My second question was, how and what do you use to solve this?
 A: In order for the circle to be inside or just touching the edge of a hexagon, the centre of the circle must be inside or on a concentric hexagon whose sides are oriented parallel to the gridlines.
The perpendicular distance from the centre of a side $36$ hexagon to an edge is $18\sqrt{3}$, so subtracting the radius of the circle, the corresponding distance from the centre to an edge of the inner hexagon is $12\sqrt{3}$.
The linear ratio between these similar shapes is therefore $2:3$ and the area ratio is therefore $4:9$.
Thus the probability you seek is the ratio of these areas, i.e. $$\frac 49$$
A: You are randomly choosing a point and then drawing a circle around it. Another way of putting it: "Throw a dart at the sheet of hexagons. Draw a circle around it. What's the probability the circle fits inside one of the hexagons?"
How to do this problem? Here are some hints:


*

*Remember that a circle simply denotes all points equidistant from the center. So another way of phrasing this is, "What is the probability that a randomly drawn point from this infinite sheet of hexagons is a distance of at least $6\sqrt{3}$ from the boundary of a hexagon?"

*Since hexagons tile the plane, your point will almost always lie within the interior of a hexagon. So it might be easier to start by contemplating a single hexagon and then generalizing from there.

*Infinities are tricky. Depending on your level of mathematical experience, you may need to formalize this question. How can you make sense of this probability on an infinite sheet?
A: Hint:
When the disk is thrown, its center point will fall inside some hexagon. You have to check the probability that the disk wholly fits, i.e. that the center is at a distance at least $r$ from all sides. The locus of such points is obtained by drawing straight lines parallel to the sides at distance $r$, which define a smaller hexagon. The probability is given by the ratio of the areas.
