In the standard notation, the area of the euclidean triangle ABC can be written as
If we recall our earlier notation for a hyperbolic triangle ABC  sinh(b)sinh(c)sin(A) = sinh(a)sinh(b)sin(C) = sinh(a)sinh(c)sin(B) = Δ, where Δ is the positive root of 1  α^{2}  β^{2}  γ^{2} + 2αβγ.
Thus the expression sinh(x)sinh(y)sin(Z), where x, y are the hyperbolic lengths of the


The quantity Δ also appears when we investigate the hyperbolic length of altitudes. Th sketch on the right shows a hyperbolic triangle ABC and the altitude AD at the vertex A. The triangle ABD has a right angle at D, so we an apply the sine formula to get sin(B) = sinh(h_{A})/sinh(c), where h_{A} = d(A,D), the length of the altitude. From the formulae above, Δ = sinh(a)sinh(c)sin(B), so we then obtain Δ = sinh(a)sinh(h_{A}). Again by the symmetry of Δ in a,b,c we get the result sinh(a)sinh(h_{A}) = sinh(b)sinh(h_{B}) = sinh(c)sinh(h_{C}) = Δ. We also have expressions for the length of the altitudes e.g. sinh(h_{A}) = Δ/sinh(a).
Now we have six expressions for Δ, each of which is an analogue of a euclidean when we try to prove a hyperbolic version of a euclidean result, then
(1) euclidean lengths are replaced by hyperbolic functions of the hyperbolic lengths,
We offer no serious attempt to justify these phenomena. We merely observe that

