Let Euler count!
A convex polyhedron has faces, of which are triangles, of which are quadrilaterals, and of which are pentagons. How many vertices does have?
The answer is 35.
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1 solution
The number of vertices V , faces F , and edges E in a convex 3-dimensional polyhedron, satisfy V + F − E = 2 . This fact is known as Euler's formula.
In this case F = 4 7 and E = 2 1 ( 3 5 × 3 + 5 × 4 + 7 × 5 ) = 8 0 so V = 2 + 8 0 − 4 7 = 3 5 .