Problem 37

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Problem 37: (2025 AMC 10b, 12)

The figure below shows an equilateral triangle, a rhombus with a $60^\circ$ angle, and a regular hexagon, each of them containing some mutually tangent congruent disks. Let $T, R,$ and $H,$ respectively, denote the ratio in each case of the total area of the disks to the area of the enclosing polygon. Which of the following is true? $\textbf{(A)}\ T=H=R \qquad$ $ \textbf{(B)}\ H<R=T \qquad$ $ \textbf{(C)}\ H=R<T\qquad$ $ \textbf{(D)}\ H<R<T \qquad$ $ \textbf{(E)}\ H<T<R \qquad$

Solution

Exercise 37: Solution

The figure below shows an equilateral triangle, a square, and a regular hexagon, each of them containing some mutually tangent congruent disks. Let $T, S,$ and $H,$ respectively, denote the ratio in each case of the total area of the disks to the area of the enclosing polygon. Which of the following is true? $\textbf{(A)}\ T=H=S\qquad$ $\textbf{(B)}\ H<S=T\qquad$ $\textbf{(C)}\ H=S<T\qquad$ $\textbf{(D)}\ H<S<T\qquad$ $\textbf{(E)}\ H<T<S$

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