2025 AMC 10B Problem 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 \lt R = T \qquad \textbf{(C)} H = R \lt T \qquad \textbf{(D)} H \lt R \lt T \qquad \textbf{(E)} H \lt T \lt R

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.