  Given a set of points in the plane, we show that the $\theta$-graph with 5
cones is a geometric spanner with spanning ratio at most $\sqrt{50 + 22
\sqrt{5}} \approx 9.960$. This is the first constant upper bound on the
spanning ratio of this graph. The upper bound uses a constructive argument that
gives a (possibly self-intersecting) path between any two vertices, of length
at most $\sqrt{50 + 22 \sqrt{5}}$ times the Euclidean distance between the
vertices. We also give a lower bound on the spanning ratio of
$\frac{1}{2}(11\sqrt{5} -17) \approx 3.798$.
