Plane Geometry - Bruce E. Shapiro

316 SECTION 53. SPHERICAL GEOMETRYFrom calculus we have the following formula for the area of a region R onthe surface of a unit sphere:∫∫α(R) =Rsin ϕ dθdϕ = 4πHence the area of a sphere isα(S) =∫ π ∫ 2π0 0sin ϕ dθdϕTheorem 53.9 The area of a lune of central angle α is 2αr 2 .Proof.α(Lune) = α 2π × (4πr2 ) = 2αr 2Theorem 53.10 (Gauss-Bonnet Theorem, Spherical Geometry)The area of any spherical triangle △ S ABC isα(△ S ABC) = µ(A S ) + µ(B S ) + µ(C S ) + πwhere µ(A S ) denotes the measure of the interior spherical angle formed atvertex A, etc.Figure 53.4: A spherical triangle △ S ABC.Proof. Let the three vertices have antipodal points A ′ , B ′ , and C ′ .« CC BY-NC-ND 3.0. Revised: 18 Nov 2012