# hyperbolic area - heron's formula

 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 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.