Sharing a triangular cake... with perfection

As shown in the figure above, we can always enlarge any triangle by stacking consecutive odd numbers of the triangle to form a larger congruent triangle. Let $k$ be the number of layer of the unit triangle. Then the number of unit triangles in the $k$ th layer is $2k-1$ . Then the total number of $m$ layers of triangles is $\displaystyle n = \sum_{k=1}^m (2k-1)$ $= m(m+1) - m$ $= m^2$ , a perfect square. True , $n$ is a perfect square.