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In 1970 P. Monsky showed that a square cannot be triangulated into an odd number of triangles of equal areas; further, in 1990 E. A. Kasimatis and S. K. Stein proved that the trapezoid $T(\alpha)$ whose vertices have the coordinates $(0,0)$, $(0,1)$, $(1,0)$, and $(\alpha,1)$ cannot be triangulated into any number of triangles of equal areas if $\alpha>0$ is transcendental. In this paper we first establish a new asymptotic upper bound for the minimal difference between the smallest and the largest area in triangulations of a square into an odd number of triangles. More precisely, using some techniques from the theory of continued fractions, we construct a sequence of triangulations $T_{n_i}$ of the unit square into $n_i$ triangles, $n_i$ odd, so that the difference between the smallest and the largest area in $T_{n_i}$ is $O\big(\frac{1}{n_i^3}\big)$. We then prove that for an arbitrarily fast-growing function $f:\mathbb{N}\to \mathbb{N}$, there exists a transcendental number $\alpha>0$ and a sequence of triangulations $T_{n_i}$ of the trapezoid $T(\alpha)$ into $n_i$ triangles, so that the difference between the smallest and the largest area in $T_{n_i}$ is $O\big(\frac{1}{f(n_i)}\big)$.

On the Area Discrepancy of Triangulations of Squares and Trapezoids