An infinte plane is tiled with equilateral triangles of side length 9cm. If you drop a coin of radius 1 cm onto the plane, what is the probability the coin lands within one of the triangles?

Answer 1

"within" is taken to mean "not touching a side"

the center of the coin must land inside of a "nested" inner triangle
whose edges are parallel to the 9 cm triangle and .5 cm inside

the ratio of the areas of the inner and outer triangles is the probability

Answer 2

the side length of the inner triangle is ... 9 - √3

the ratio of the areas is the square of the ratio of the side lengths