ON HYPERBOLIC AREA OF THE MODULI OF θ－ACUTE TRIANGLES

A θ-acute triangle is a Euclidean triangle on the plane whose three angles are less than a given constant θ. In this note, we shall give an explicit formula computing the hyperbolic area A(θ) of the moduli region of θ-acute triangles on the Poincar´e disk. It turns out that A(θ) is a period in the sense of Kontsevich-Zagier if cot θ is a nonnegative algebraic number.