jmc

Let $O=(0,0)$, and let $A$ and $B$ denote the points where $y=6$ intersects $y=x$ and $y=-x,$ respectively. The coordinates of $A$ and $B$ are $(6,6)$ and $(-6,6),$ respectively, so $AB=6-(-6)=12.$ Also, the length of the altitude from $O$ to $AB$ is $6.$ Thus the area of $△OAB$ is$$\frac{1}{2}\cdot 12\cdot 6=\boxed{36}.$$