An equilateral triangle is constructed externally on each side of a square with side length $2$. What is the area of the resulting octagon?

1. The square has area $2^2=4$. Each of the four equilateral triangles has area $\frac{\sqrt3}{4}(2^2)=\sqrt3$.
2. The resulting octagon has area $4+4\sqrt3=\boxed{4+4\sqrt3}$.