Euler's formula for the pentagonal prism

Statement

Verify Euler’s formula for a prism with a pentagonal base.

Solution

Let us count the elements of the pentagonal prism:

• Faces: $2$ pentagonal bases $+ 5$ lateral faces $= 7$, hence $F=7$.
• Vertices: $5$ on the bottom base $+ 5$ on the top one $= 10$, hence $V=10$.
• Edges: $5+5$ of the two bases $+ 5$ vertical ones $= 15$, hence $S=15$.

Then $V - S + F = 10 - 15 + 7 = \boxed{2}. \quad ✓$

Links

Topics: Synthetic geometry in space
Concepts: Euler’s formula · Polyhedron · Prism
Skills: Proving · Synthetic geometry
Exercise types: Geometric problem