Describe the geodesics in the disc model of the hyperbolic plane $\mathbb{H}^{2}$.

Define the area of a region in $\mathbb{H}^{2}$. Compute the area $A(r)$ of a hyperbolic circle of radius $r$ from the definition just given. Compute the circumference $C(r)$ of a hyperbolic circle of radius $r$, and check explicitly that $d A(r) / d r=C(r)$.

How could you define $\pi$ geometrically if you lived in $\mathbb{H}^{2}$ ? Briefly justify your answer.