Ever Increasing Trapezia

Beautiful Problem.

Here is how I did it.

It's not hard to verify that for every $n$ (with $1\leq n \leq 26$ ), the union of regions labelled by the first $n$ letter(s) of alphabet is an equilateral triangle, with side length $T_n$ , $n$ th triangular number= $\dfrac{n(n+1)}{2}$ .

Then the area of $R_{24}$

= The area of the union of regions labelled by the first $24$ letter(s) of alphabet minus The area of the union of regions labelled by the first $23$ letter(s) of alphabet

= $\dfrac{\sqrt{3}}{4} T_{24}^2 - \dfrac{\sqrt{3}}{4} T_{23}^2$

= $\dfrac{\sqrt{3}}{4}( T_{24}^2-T_{23}^2)$

= $\dfrac{\sqrt{3}}{4}( 300^2 - 276^2)$

= $3456 \sqrt{3}$

So, $a=\boxed{3456}$ .