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Problem 724

AIME late

Geometry Difficulty 5.3 Find the answer

7 Coins of the same size are arranged on a very large table (the infinite plane) such that each coin touches six other coins. Find the percentage of the plane that is covered by the coins.
(A) $\frac{20}{\sqrt{3}} \pi \%$
(B) $\frac{50}{\sqrt{3}} \pi \%$
(C) $16 \sqrt{3} \pi \%$
(D) $17 \sqrt{3} \pi \%$
(E) $18 \sqrt{3} \pi \%$

Your answer

Official solution

7 Answer: (B)
Consider the equilateral triangle formed by joining the centres of 3 adjacent coins. It is easy to see that the required percentage is given by the percentage of the triangle covered by these three coins.
We have $\frac{\frac{1}{2} \pi r^{2}}{\sqrt{3} r^{2}} \times 100 \%=\frac{50}{\sqrt{3}} \pi \%$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.