Sum of interior angles of the faces of polyhedron

Geometry Level pending

Suppose that a convex polyhedron has V V vertices, E E edges and F F faces.

What is the sum of interior angles of the faces of polyhedron (in radians)?

( E − 2 ) π (E-2)\pi 2 ( V − 2 ) π 2(V-2)\pi ( V + F − 4 ) π (V+F-4)\pi ( F − 2 ) π (F-2)\pi

Alice Smith by Alice Smith , 20, China

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1 solution

Mark Hennings
Oct 14, 2019

Let F k F_k be the number of faces that are k k -gons. Then the internal angles of all the faces sum to ∑ k π ( k − 2 ) F k = π ∑ k k F k − 2 π ∑ k F k = 2 π E − 2 π F = 2 π ( E − F ) = 2 π ( V − 2 ) \sum_k \pi(k-2)F_k = \pi\sum_k k F_k - 2\pi\sum_k F_k = 2\pi E- 2\pi F = 2\pi(E-F) =2 \pi(V-2) using Euler’s Theorem, since ∑ k k F k \sum_k k F_k counts every edge twice, so equals 2 E 2E .

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