Brilliantian Graphing System

It was recently discovered that the citizens of the lost city of Brilliantia developed a unique graphing system in which the $x$ -, $y$ -, and $z$ -axes were placed at $60°$ intervals on a two-dimensional plane. To graph an $(x, y, z)$ point, the Brilliantians would start at the origin and measure $x$ units in the in the positive $x$ direction, then from that point measure out $y$ units in the positive $y$ direction, and then from that point measure out $z$ units in the positive $z$ direction. The point $(2, 3, -1)$ in the Brilliantian system is shown below:

If the area of the region defined by $x + y + z = 10$ in the Brilliantian system for positive real values of $x$ , $y$ , and $z$ can be expressed as $a\sqrt{b}$ , where $a$ is an integer and $b$ is a square-free integer, find $a + b$ .