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?

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. $$\frac 49$$