1972 AMC 12 Problem 35

Track the orientation after one eight-pivot circuit of the square

Three circuits are needed; classify the pivots according to whether $P$ is fixed or moves through a $120^\circ$ or $30^\circ$ arc

One circuit of the square uses $8$ pivots and changes the triangle’s orientation by $\frac{2}{3}$ of a full turn. Therefore $3$ circuits, or $24$ pivots, are required to restore the entire triangle. In $8$ of those pivots the rotation is about $P,$ so $P$ does not move. In the other $16,$ eight arcs subtend $120^\circ$ and eight subtend $30^\circ,$ all with radius $2.$ Hence the total path length is $$\begin{aligned} &8\left(\frac13\cdot4\pi\right) +8\left(\frac1{12}\cdot4\pi\right)\\ &\qquad=\frac{32\pi}{3}+\frac{8\pi}{3}\\ &\qquad=\frac{40\pi}{3}. \end{aligned}$$

Therefore, the correct answer is D.