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King's Maths School

Challenge 471: Circles Within Circles

What is the probability that I can land a coin inside these shapes?

(1)

I draw an (infinite) grid of squares with side length 4 cm. I flip a circular coin of radius 1 cm onto this grid. What is the probability that the coin lands completely within one of the squares, i.e. not touching any of the lines I have drawn?

(2) [This is a fair bit harder, and unusually I would recommend using a calculator]

I draw (infinitely many) circles of radius 4 so that each circle is tangent to 4 other circles. I flip a circular coin of radius 1 onto this. What is the probability that the coin lands so that it is not touching any of the lines I have drawn?

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