Question

Assuming a polyhedron has 4 edges, 5 faces, and 3 vertices. Does such a polyhedron exist?

• A. yes
• B. no

Answer 1

The correct option is B no

Given, number of edges $\left(E\right)=4,$
number of faces $\left(F\right)=5$
and number of vertices $\left(V\right)=3$

The Euler formula gives the relation between the number of faces$\left(F\right),$ vertices$\left(V\right)$ and edges$\left(E\right)$ of any polyhedron, which is given by:
$F+V-E=2$

Let's find $F+V-E:$
$F+V-E=5+3-4$
$=8-4$
$=4$
$\ne 2$

Clearly, the given polyhedron does not satisfy the Euler's formula, therefore, a polyhedron with the given dimensions does not exist.

Option (b.) is the correct choice.