Faces, edges, vertices, prisms and pyramids

A polyhedron is a solid whose boundary is made entirely of flat polygon surfaces, with no curved parts and no holes through it.

• A face is one flat polygon surface of the solid.
• An edge is the straight line segment along which exactly two faces meet.
• A vertex (plural vertices) is a corner point where three or more edges come together.

Counting these three numbers on many different solids — cube, cuboid, triangular prism, square pyramid — reveals a fixed relationship between them, known as Euler's formula:

where: $F$ is the number of faces, $V$ is the number of vertices, and $E$ is the number of edges of the polyhedron. It says that faces plus vertices always beats edges by exactly two, for any simple polyhedron whatsoever — cube, prism, pyramid, or an oddly shaped crystal. At this level Euler's formula is something you verify by counting a solid's F, V, E and checking the sum, not something you are asked to prove in general.

Prisms. A prism has two identical, parallel polygonal faces called bases, joined by rectangular (or parallelogram) side faces. It takes its name from its base shape — a triangular prism has triangular bases, a hexagonal prism has hexagonal bases. For a prism whose base is an $n$-sided polygon, the three counts are $F = n + 2$, $V = 2n$, and $E = 3n$.

Pyramids. A pyramid has one polygonal base and triangular side faces that all meet at a single top point, the apex. It too is named after its base — a square pyramid has a square base; a triangular pyramid, called a tetrahedron when every one of its four faces is a triangle, has a triangular base. For a pyramid whose base is an $n$-sided polygon, the counts are $F = n + 1$, $V = n + 1$, and $E = 2n$.

Question. A pyramid has a hexagonal base ($n = 6$). Find its number of faces, vertices, and edges, verify Euler's formula, and then find the base shape of a different pyramid that is known to have $9$ edges.

Step 1. Faces of the hexagonal pyramid: $F = n + 1 = 6 + 1 = 7$, made of $1$ hexagonal base plus $6$ triangular side faces.

Step 2. Vertices: $V = n + 1 = 6 + 1 = 7$, made of $6$ base corners plus $1$ apex.

Step 3. Edges: $E = 2n = 2 \times 6 = 12$, made of $6$ base edges plus $6$ edges running up to the apex.

Step 4. Check: $F + V = 7 + 7 = 14$, and $E + 2 = 12 + 2 = 14$ — they match, so Euler's formula holds for this pyramid.

Answer: the hexagonal pyramid has $F = 7$, $V = 7$, $E = 12$.

Competitive stretch

Step 1. For the second pyramid, use $E = 2n$ with $E = 9$: since $9$ is odd and $2n$ is always even, no whole-number pyramid has exactly $9$ edges — the question is a trap testing whether you check that $n$ must come out as a positive whole number.

Step 2. If instead the pyramid had $E = 10$ edges, then $2n = 10$ gives $n = 5$, a pentagonal pyramid, with $F = n+1 = 6$ and $V = n+1 = 6$; check: $6 + 6 = 12 = 10 + 2$. Answer: a pyramid cannot have 9 edges, since edge counts of pyramids are always even; 10 edges gives a pentagonal pyramid.

• Mixing up prisms and pyramids — a prism has two parallel bases and rectangular side faces, a pyramid has one base and triangular side faces meeting at a single apex; using the wrong formula set gives a wrong count immediately.
• Forgetting edges hidden at the back of a 3-D drawing — every edge counts even if it is not directly visible on the page.
• Quoting Euler's formula from memory without actually counting F, V, E on the solid first — always count, then check the formula holds, rather than assuming it.
• Forgetting that a pyramid's edge count $E = 2n$ is always even, so an odd number can never be a valid edge count for any pyramid.

Ask the tutor: Want to practice counting faces, edges, and vertices on a trickier solid, or work through more reverse-lookup problems using Euler's formula? Ask the AI tutor to walk through extra examples with you.