Question
Verify Euler’s Formula for a cube:
where = number of faces, = number of vertices, = number of edges.
Solution — Step by Step
A cube has 6 flat square faces — top, bottom, front, back, left, right. So .
A vertex is a corner point where edges meet. A cube has 8 corners (think: 4 on the top face, 4 on the bottom face). So .
An edge is a line segment where two faces meet. A cube has 12 edges — 4 on the top face, 4 on the bottom face, and 4 vertical edges connecting them. So .
Substitute into the formula:
The formula holds. Euler’s Formula is verified for a cube.
Why This Works
Euler’s Formula — — is a fundamental relationship in topology that holds for any convex polyhedron (a 3D solid with flat faces and no holes). Leonhard Euler discovered it in the 18th century, and it’s truly remarkable that this single equation works for a tetrahedron, a cube, a dodecahedron — all of them give exactly 2.
The intuition: as you add faces to a polyhedron, you always add vertices and edges in a balanced way such that the formula stays constant. It’s not a coincidence — it reflects a deep structural property of 3D solids.
For CBSE Class 8, we just need to verify it for standard shapes. But remember: the formula only works for polyhedra — not for shapes with holes or curved surfaces.
Alternative Method
You can also think of it as building the cube step-by-step and checking at each stage.
Start with one square face: , , . Check: .
Now “fold up” the cube by adding the remaining 5 faces one at a time. Each new face adds 1 to , adds some edges and vertices, and the count always adjusts to keep the formula true at the final closed shape.
This is more of a conceptual walkthrough than a calculation method, but it helps you see why the numbers balance out rather than just checking them mechanically.
Common Mistake
The most common error is miscounting edges — students often count 24 edges instead of 12. This happens when you count each edge twice (once for each face it belongs to). Remember: an edge is shared between two faces. Each of the 12 edges belongs to exactly 2 faces, so if you’re listing by face (6 faces × 4 edges = 24), you must divide by 2. The cube has 12 edges, not 24.
A quick memory trick: 6, 8, 12 — these are the F, V, E values for a cube. Notice and . You can also remember that 12 edges = 4 top + 4 bottom + 4 vertical. This breakdown makes counting foolproof in an exam.