In euclidean geometry, a triangle, and hence its area, is determined by the lengths of
its sides. The formula which gives the area in terms of the lengths is
heron's formula
If a triangle has sides of lengths a, b and c, then its area D is given by
D^{2} = s(sa)sb)(sc)),
where s = (a+b+c)/2  the semiperimeter.
In hyperbolic geometry also, the (hyperbolic) lengths of the sides determine a triangle.
We shall establish the analogue of heron's formula. Note that, in hyperbolic geometry
a triangle is determined by its angles. The formula relating the area to the angles is,
of course, the gaussbonnet formula!
heron's formula for hyperbolic triangles
If a hyperbolic triangle has sides of lengths a, b and c, then its area D is given by
cos(D) = (α+β+γ+αβ+βγ+γα+α^{2}+β^{2}+γ^{2}
αβγ)/(1+α)(1+β)(1+γ), and hence by
tan(D/2) = Δ/(1+α+β+γ),
where α=cosh(a),β=cosh(b),γ=cosh(c), and
Δ^{2} = 1α^{2}β^{2}γ^{2}+2αβγ, Δ > 0.
Also, sin(D/2) = Δ/4cosh(½a)cosh(½b)cosh(½c)
Note that, from the gaussbonnet formula, Dε(0,π), so any one determines D uniquely.
Since it involes the sum α+β+γ we may regard the latter as the better analogue.
proof
As in euclidean geometry, we get simper results for rightangled triangles.
area of a rightangled hyperbolic triangle
If the hyperbolic triangle ABC has a right angle at A, and
d(A,B)=c, d(B,C)=a, d(C,A)=b, then its hyperbolic area D
is given by
sin(D) = sinh(b)sinh(c)/(cosh(a)+1).
proof
We leave it as an exercise to show that we also have
cos(D) = (cosh(b)+cosh(c)/(cosh(a)+1).
alternative formula for the area of a rightangled hyperbolic triangle
If the hyperbolic triangle ABC has a right angle at C, then its area D is given by
tan(D/2) = tanh(a/2)tanh(b/2)
Although it is quite easy to prove this directly, we may as well use heron's formula.
Since the triangle is rightangled at C, cosh(c) = cosh(a)cosh(b), i.e. γ = αβ.
Then Δ^{2} = 1α^{2}β^{2}α^{2}β^{2}+2α^{2}β^{2}
= (α^{2}1)(β^{2}1) = (sinh(a)sinh(b))^{2}.
Also, 1+α+β+γ = 1+α+β+αβ = (α+1)(β+1) = (cosh(a)+1)(cosh(b)+1).
Now, heron's formula gives tan(D/2) = (sinh(a)/(cosh(a)+1)).(sinh(b)/(cosh(b)+1)).
The result follows as sinh(x)/(cosh(x)+1) = tanh(x/2).
This last result is useful in determining the hyperbolic area of a hyperbolic circle.
