Euler's Formula is a fundamental relationship in geometry that connects the number of vertices (V), edges (E), and faces (F) of a convex polyhedron. A polyhedron is a 3-dimensional solid with flat polygonal faces. For any convex polyhedron, the following formula holds true:

$V - E + F = 2$

Let's break this down:

Step 1: Understand the Terms
- Vertices (V): The corner points where edges meet.
- Edges (E): The line segments where two faces meet.
- Faces (F): The flat surfaces that make up the polyhedron.

Step 2: Verify with an Example
Consider a cube:
- It has 8 vertices.
- It has 12 edges.
- It has 6 faces.
Plug into Euler's Formula: $8 - 12 + 6 = 2$ . It holds true.

Step 3: The Correct Option
Comparing with the given options, the correct formula is $V - E + F = 2$ , which corresponds to the option: $V - F + E = 2$ (since addition is commutative, V - F + E is the same as V + E - F, but note the standard form is V - E + F = 2. The option $V - F + E = 2$ is mathematically equivalent to V + E - F = 2, which is not the standard Euler's formula. The correct standard form is V - E + F = 2. Among the options, the one that matches this is actually $V + F - E = 2$ , because V + F - E = V - E + F. So the correct choice is $V + F - E = 2$ .

Final Answer: The correct Euler's formula is $V + F - E = 2$ .

Related Topics and Formulae

Euler's Formula: This formula applies to any convex polyhedron. It is a topological invariant, meaning it depends on the shape's overall structure rather than its specific geometry. It is also used in graph theory for planar graphs, where V - E + F = 1 + C, with C being the number of connected components.

Key Properties: The formula highlights a deep connection between the combinatorial properties of polyhedra. It is independent of the polyhedron's size or shape, as long as it is convex and has no holes (genus 0).