## Abstract

We begin this chapter by defining and computing the area of a sphere and establishing a famous result of Girard on the area of a *spherical triangle*. Then, we present the important concept of *convex polyhedron*, which encompasses prisms and pyramids, and apply Girard’s theorem to prove the celebrated *theorem of Euler*, which asserts that the *Euler characteristic* of every convex polyhedron is equal to 2. The chapter finishes with using Euler’s theorem to obtain the classification of all *regular* polyhedra, and showing that all found possibilities do exist.

## References

- 2.T. Apostol,
*Calculus*, vol. 1 (Wiley, New York, 1967)Google Scholar - 5.A. Caminha,
*An Excursion Through Elementary Mathematics I - Real Numbers and Functions*(Springer, New York, 2017)Google Scholar

## Copyright information

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