Dual of the formula for Euclidean triangles It’s well

meli199939f

meli199939f

Answered question

2022-04-04

Dual of the formula for Euclidean triangles
It’s well known that the trigonometric area formula for Euclidean triangles is
S=12absinC (1.1)
Is there a such formula for hyperbolic triangles?
Is there a proof for (1.1) by using law of cosines?

Answer & Explanation

Esteban Sloan

Esteban Sloan

Beginner2022-04-05Added 21 answers

You may derive a similar expression for hyperbolic triangles by combining Heron's formula
tanhΔ4=tanhs2tanhsa2tanhsb2tanhsc2,s=a+b+c2 (1)
or the expression for the area in terms of the angular defect
Δ=π(A+B+C) (2)
together with the laws of sines and cosines:
sinhAsina=sinhBsinb=sinhCsinc,coshc=coshacoshbsinhasinhbcosC
Given a,b,C, you may derive c, then s, from the law of cosines, then plug a,b,c,s into (1).
You can do the same in the Euclidean context.

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