section_path: Role and characterizations in mathematics/Topology
section_name: Topology
paragraph_text: Topology
  
  The constant π appears in the Gauss–Bonnet formula which relates the differential geometry of surfaces to their topology. Specifically, if a compact surface Σ has Gauss curvature K, then ∫ Σ K d A = 2 π χ (Σ) {\displaystyle \int _{\Sigma }K\,dA=2\pi \chi (\Sigma)} where χ (Σ) is the Euler characteristic, which is an integer. An example is the surface area of a sphere S of curvature 1 (so that its radius of curvature, which coincides with its radius, is also 1.) The Euler characteristic of a sphere can be computed from its homology groups and is found to be equal to two. Thus we have A (S) = ∫ S 1 d A = 2 π ⋅ 2 = 4 π {\displaystyle A(S)=\int _{S}1\,dA=2\pi \cdot 2=4\pi } reproducing the formula for the surface area of a sphere of radius 1.
  
  The constant appears in many other integral formulae in topology, in particular, those involving characteristic classes via the Chern–Weil homomorphism.