Sitting spaces

Let two square mats be placed with one corner of one in contact with one corner of another. It is tempting to surround the in between place for a game, a deity, a yagya or just leave vacant by placing a square mat at $PQ$.

Sitting Space 1

The sitting space $S$ provided by the third mat as shown in Figure $\ref{sit-1}$ is not as much as provided by the earlier mats. If we increase the angle $\theta$ in between the two mats we started with, the space $S$ is seen to increase.
                                
 

Sitting Space 2

Let us vary the angle $\theta$, and study the corresponding changes on the space S. Observe situations:
$$S < A,\ \ S > A.$$
From the situations $S < A$, can it suddenly turn to situations $S > A$?

Must it not pass through the situation $S = A$?

What is the corresponding value of $\theta$?

Is it not interesting to experiment that as we increase $\theta$ further, up goes $S$ (above $A$).  Continue the experiment  by increasing $\theta$ and observing the increase in $S$. What is the maximum possible value of $\theta$, and what is the corresponding value of $S$, in terms of $A$?

It is interesting to discover that you cross the situations $S > 2A$, $S > 3A,\, \ldots$.

What happens as long as $\theta$ is acute; and what happens when it turns obtuse? Mark the turn from situations $S < 2A$ to $S > 2A$; can we characterise them in terms of angle $\theta$ (acute and obtuse values of $\theta$ ). In between must be the stage $S = 2A$. What is $\theta$ then? Formulate your discovery as a theorem.

Let us repeat the experiment with the hands of a clock, attaching stiff paper to hands making them sides of two squares with spaces $A$ and $B$ in between varying angle $\theta$. Discuss situations for acute and obtuse values of $\theta$.

                                                             Clock hands
Can situations $S < A+B$ suddenly turn to $S > A+B$?  When is $S = A+B$? Formulate your finding in the form of a theorem.  Investigate whether your theorem remains true in case, the sitting mats we start with are semicircular or triangular (equilateral) instead of being square.