Counting a Hexagonal Prism Systematically

Count the vertices, edges and faces of a hexagonal prism and check Euler's formula.

Count the vertices layer by layer: The top hexagon has $6$ corners and the bottom hexagon has $6$ corners = $V = 6 + 6 = 12$

Count the edges in groups: $6$ edges around the top, $6$ around the bottom, $6$ vertical edges joining them = $E = 6 + 6 + 6 = 18$

Count the faces: $2$ hexagonal bases and $6$ rectangular side faces = $F = 2 + 6 = 8$

Check Euler's formula: $V - E + F = 12 - 18 + 8$ = $2$

Answer: $V = 12$, $E = 18$, $F = 8$ and $12 - 18 + 8 = 2$, so Euler's formula holds.

Counting Edges from the Faces

A polyhedron has $8$ triangular faces and $6$ square faces. How many edges and how many vertices does it have?

Add up the sides of all faces: $8 \times 3 + 6 \times 4 = 24 + 24$ = $48$ sides

Each edge is shared by two faces, so divide by 2: $E = 48 \div 2$ = $E = 24$

Count the faces: $F = 8 + 6$ = $F = 14$

Solve Euler's formula for $V$: $V = 2 + E - F = 2 + 24 - 14$ = $V = 12$

Answer: The polyhedron has $24$ edges and $12$ vertices.

Can This Solid Exist?

Someone claims to have built a convex polyhedron with $10$ vertices, $16$ edges and $7$ faces. Is that possible?

Substitute into $V - E + F$: $10 - 16 + 7$ = $1$

Compare with Euler's formula: A convex polyhedron must give $2$, not $1$ = Impossible

Find the face count that would work: $10 - 16 + F = 2$, so $F = 2 + 16 - 10$ = $F = 8$

Answer: No. $10 - 16 + 7 = 1 \neq 2$, so the counts are wrong. With $10$ vertices and $16$ edges a convex polyhedron would need $8$ faces.

A Prism Known Only by Its Edges

A prism has $36$ edges. How many vertices and faces does it have? What shape is its base?

Use $E = 3n$ for a prism: $3n = 36$ = $n = 12$, the base is a 12-gon

Find the vertices with $V = 2n$: $V = 2 \times 12$ = $V = 24$

Find the faces with $F = n + 2$: $F = 12 + 2$ = $F = 14$

Check with Euler's formula: $24 - 36 + 14$ = $2$

Answer: The base is a 12-gon (dodecagon). The prism has $24$ vertices and $14$ faces, and $24 - 36 + 14 = 2$.

A Non-Example: a Solid with a Hole

A square picture frame is a solid with a square hole through the middle. Split its top and bottom into $4$ trapezoids each, so every face is a polygon. Count $V$, $E$ and $F$ and compute $V - E + F$.

Count the vertices: Top: $4$ outer + $4$ inner corners. Bottom: the same again = $V = 16$

Count the faces: $4$ trapezoids on top, $4$ on the bottom, $4$ outer sides, $4$ inner sides = $F = 16$

Count the edges from the faces: All $16$ faces are quadrilaterals: $16 \times 4 \div 2$ = $E = 32$

Compute $V - E + F$: $16 - 32 + 16$ = $0$

Answer: $V - E + F = 0$, not $2$. Euler's formula $V - E + F = 2$ only holds for solids without holes; each hole lowers the value by $2$.

Mistake: Adding up the sides of all faces and calling that the number of edges

Why: Every edge is the meeting line of two faces, so it is counted once for each of them.

Correct: Divide the total number of face sides by 2. A cube has $6 \times 4 = 24$ face sides but only $24 \div 2 = 12$ edges.

Mistake: Mixing up the signs, for example writing $V + E - F = 2$

Why: The edges are the count that is subtracted. With the wrong signs a cube gives $8 + 12 - 6 = 14$.

Correct: Vertices and faces are added, edges are subtracted: $V - E + F = 2$. Check it on a cube: $8 - 12 + 6 = 2$.

Mistake: Using the formula on cylinders, cones, spheres or solids with holes

Why: Curved surfaces are not polygons, and a hole changes the result (a picture frame gives $0$).

Correct: Use $V - E + F = 2$ only for polyhedra without holes: flat polygon faces, straight edges, one closed surface.

Mistake: Thinking that any three numbers with $V - E + F = 2$ describe a real polyhedron

Why: The formula is a test that every polyhedron passes, but passing it is not enough. For example $V = 4.5$, $E = 7.5$, $F = 5$ also gives $2$.

Correct: Also check that the counts are whole numbers and that the face sides add up to an even number ($2E$). Five triangles have $15$ sides, which is odd, so no polyhedron is made of exactly five triangles (a polyhedron always has an even number of odd-sided faces).

Designing a Soccer Ball

The classic soccer ball is stitched from $12$ pentagons and $20$ hexagons. Designers use face sides and Euler's formula to know how many seams and seam corners there are before cutting any material.

Each seam joins two panels, so the number of seams is half the number of panel sides.

3D Models in Games and Animation

Characters and objects in video games are built from closed meshes of triangles. Software checks Euler's formula to catch meshes with gaps, duplicated corners or accidental holes before they cause glitches.

A closed mesh made of $4$ triangles (a tetrahedron) has $6$ edges and $4$ vertices: $4 - 6 + 4 = 2$.

Crystals in Minerals

Many crystals grow as polyhedra. A quartz crystal attached to a rock is shaped like a hexagonal prism with a hexagonal pyramid on its free end, and mineralogists use counts like these to describe crystal forms.

A quartz point has $6$ rectangular sides, $6$ triangular faces at the tip and $1$ hexagonal face where it sits on the rock.

Q: Does Euler's formula work for polyhedra that are not convex?

A: Yes, as long as the solid has no holes (it could be inflated like a balloon into a sphere). An L-shaped block, for example, still gives $V - E + F = 2$. Solids with holes, like a picture frame, give a different value.

Q: Why is the answer always 2?

A: Imagine removing one face and flattening the rest onto a table. Splitting the faces into triangles and then removing outer triangles one at a time never changes $V - E + F$, and you end with a single triangle ($3 - 3 + 1 = 1$). Adding back the removed face gives $1 + 1 = 2$. The number $2$ belongs to the sphere-like shape of the surface, not to the particular solid.

Q: Does the formula work for a cylinder or a cone?

A: No. They have curved surfaces, so they are not polyhedra and their faces and edges are not counted the same way. Euler's formula is about solids made only of flat polygons.

Q: Who discovered the formula?

A: Leonhard Euler described it in a letter in 1750 and published it in the following years. René Descartes had written down a closely related result around 1630, but his notes were only found much later.