Extended Hyperbolic Surfaces in $R^3$

In this paper, I will describe the construction of several surfaces whose intrinsic geometry is hyperbolic geometry, in the same sense that spherical geometry is the geometry of the standard sphere in Euclidean 3-space. I will prove that the intrinsic geometry of these surfaces is, in fact, (a close approximation of) hyperbolic geometry. I will share how I (and others) have used these surfaces to increase our own (and our students') experiential understanding of hyperbolic geometry. (How to find hyperbolic geodesics? What are horocycles? Does a hyperbolic plane have a radius? Where does the area formula $\pi r^2$ fit in hyperbolic geometry?).