Question

求证:有V个顶点的凸多面体的各面多边形的内角总和为(V-2)·360°.

Answer 1

证明:V+F-E=2,E-F=V-2.

设凸多面体的各面分别是E₁、E₂、…、E_F边形,则各面多边形内角总和是

(E₁-2)·180°+(E₂-2)·180°+…+(E_F-2)·180°

=(E₁+E₂+…+E_F-2F)·180°

=(2E-2F)·180°=(E-F)·360°

=(V-2)·360°.