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
If a triangle has sides of lengths a, b and c, then its area D is given by
D2 = s(s-a)s-b)(s-c)),
where s = (a+b+c)/2 - the semi-perimeter.
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 gauss-bonnet 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+α2+β2+γ2-
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
Δ2 = 1-α2-β2-γ2+2αβγ, Δ > 0.
Also, sin(D/2) = Δ/4cosh(½a)cosh(½b)cosh(½c)
Note that, from the gauss-bonnet 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.
As in euclidean geometry, we get simper results for right-angled triangles.
area of a right-angled 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).
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 right-angled 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 right-angled at C, cosh(c) = cosh(a)cosh(b), i.e. γ = αβ.
Then Δ2 = 1-α2-β2-α2β2+2α2β2
= (α2-1)(β2-1) = (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.