On the area formulas of inscribed polygons in classical geometry (Q2214890)

In [\textit{Y. Matsumoto} et al., Enseign. Math. (2) 53, No. 1--2, 127--153 (2007; Zbl 1142.51016)] it was shown that, in Euclidean geometry, there is no area formula of the general cyclic \(n\)-gon for \(\geq 5\) in terms of its side lengths which uses only the four arithmetic operations of addition, subtraction, multiplication and division, and \(k\)-th rooots. In this paper, the authors show that a similar result holds for the hyperbolic and the spherical setting, in the precise sense that: If \(S\) is the area of a cyclic \(n\)-gon for \(n \geq 5\) in hyperbolic or spherical geometry, whose side lengths are \(a_1, a_2 , \dots, a_n\). then there is no formula for \(\cos \frac{S}{2}\) in terms of \(s(a_1)\), \(s(a_2), \dots,s(a_n)\) that uses only arithmetic operations and \(k\)-th roots, where \(s(x)\) denotes \(\sinh \frac{x}{2}\) in the case of hyperbolic geometry and \(\sin \frac{x}{2}\) in the case of spherical geometry.