  Theta graphs are important geometric graphs that have many applications,
including wireless networking, motion planning, real-time animation, and
minimum-spanning tree construction. We give closed form expressions for the
average degree of theta graphs of a homogeneous Poisson point process over the
plane. We then show that essentially the same bounds---with vanishing error
terms---hold for theta graphs of finite sets of points that are uniformly
distributed in a square. Finally, we show that the number of edges in a theta
graph of points uniformly distributed in a square is concentrated around its
expected value.
