Question

A polyhedron has 15 edges and 5 faces. Use Euler's formula to find the number of vertices.

Answer 1

Euler's formula for polyhedra states that for any convex polyhedron, the relationship between the number of vertices \( V \), edges \( E \), and faces \( F \) is given by:

\[
V - E + F = 2
\]

In this case, we are given:
- \( E = 15 \) (the number of edges)
- \( F = 5 \) (the number of faces)

We need to find the number of vertices \( V \). We can rearrange Euler's formula to solve for \( V \):

\[
V = E - F + 2
\]

Substituting in the known values:

\[
V = 15 - 5 + 2
\]

Calculating the right-hand side:

\[
V = 10 + 2 = 12
\]

Thus, the number of vertices \( V \) is:

\[
\boxed{12}
\]