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? $\textbf{(A)}\ T=H=S\qquad$ $\textbf{(B)}\ H<S=T\qquad$ $\textbf{(C)}\ H=S<T\qquad$ $\textbf{(D)}\ H<S<T\qquad$ $\textbf{(E)}\ H<T<S$

1. In each polygon let $\rho$ be the radius of the disks; the ratios do not depend on $\rho$.
2. \emph{Triangle.} Each disk touches the two sides at a vertex of angle $60^\circ$, so its center lies on the angle bisector at distance $\rho/\sin30^\circ=2\rho$ from that vertex. The three disks are tangent, so their centers form an equilateral triangle of side $2\rho$ and circumradius $\frac{2\rho}{\sqrt3}$. The circumradius of the big triangle is therefore $\frac{2\rho}{\sqrt3}+2\rho$, and its side is $\sqrt3\left(\frac{2\rho}{\sqrt3}+2\rho\right)=2\rho(1+\sqrt3)$. Its area is $\frac{\sqrt3}4\cdot4\rho^2(1+\sqrt3)^2=\rho^2(6+4\sqrt3)$, so
   $$T=\frac{3\pi\rho^2}{\rho^2(6+4\sqrt3)}=\frac{\pi(2\sqrt3-3)}2\approx0.232\,\pi.$$
3. \emph{Square.} Each disk touches two sides at a vertex of angle $90^\circ$. The four tangent disks have centers forming a square of side $2\rho$, and the sides of the enclosing square are parallel to it at distance $\rho$ outside. So the enclosing square has side $2\rho+2\rho=4\rho$ and area $16\rho^2$, and
   $$S=\frac{4\pi\rho^2}{16\rho^2}=\frac\pi4=0.25\,\pi.$$
4. \emph{Hexagon.} The six tangent disks have centers forming a regular hexagon of side $2\rho$ and circumradius $2\rho$; each vertex of it points to the midpoint of a side of the enclosing hexagon, which that disk touches. Hence the apothem of the enclosing hexagon is $2\rho+\rho=3\rho$, its side is $\frac{2}{\sqrt3}\cdot3\rho=2\sqrt3\rho$, and its area is $6\cdot\frac{\sqrt3}4\cdot12\rho^2=18\sqrt3\rho^2$. Thus
   $$H=\frac{6\pi\rho^2}{18\sqrt3\rho^2}=\frac{\pi}{3\sqrt3}\approx0.192\,\pi.$$
5. \emph{Comparison.} $T