Chapter 13.3 — Exercise 13.2 — Euler's Theorem

Congruent Faces Versus Merely Flat Ones

Among polyhedra, a second distinction separates regular from non-regular. A regular polyhedron has every face congruent to every other face — a cube qualifies, since all six of its faces are identical squares. A cuboid does not, since its three pairs of rectangular faces generally come in three different sizes. This is the same regular-versus-not distinction that separates a square from an ordinary rectangle in flat geometry, just one dimension up.

Two Families Built From a Base Shape

Most of the polyhedra in this chapter belong to one of two families, both named after whatever polygon forms their base:

The naming convention is entirely about the base: a hexagonal prism has two hexagonal ends and six rectangular sides, while a hexagonal pyramid has one hexagonal base and six triangular sides meeting at a point above it — same base shape, completely different solid.

Three Numbers Worth Counting on Any Solid

Every polyhedron can be described by three counts: its faces (F), its vertices (V) — the corner points where edges meet — and its edges (E), the line segments where two faces meet.

A cube and a cuboid share exactly the same three counts, despite looking noticeably different — a first hint that F, V, and E alone don't fully pin down a solid's shape, only some of its structure.

One Relationship, True for Every Row Above

Adding F and V together, and separately adding E and 2, produces the same total in every single row of that table: 6+8=14 and 12+2=14 for the cube; 4+4=8 and 6+2=8 for the tetrahedron; 7+7=14 and 12+2=14 for the hexagonal pyramid. This is Euler's relation, named for the mathematician Leonhard Euler, and it holds for every polyhedron without a single exception:

Applying it to seven more polyhedra confirms it holds regardless of how many faces or vertices are involved:

Every row balances exactly. Notice too that two entirely different-looking polyhedra can land on the exact same F, V, E triple (8, 12, 18 appears among these the same way it did for the hexagonal prism above) — Euler's relation doesn't identify a unique shape, but every genuine polyhedron, whatever its shape, has to satisfy it.

Using the Relation to Find a Missing Count

Once F+V=E+2 is trusted, any one of the three counts can be recovered from the other two, without needing to see the solid or count anything directly:

This is the same relation used three different ways — solved for E in the first row, for V in the second, and for F in the third — depending on which count happens to be missing.

Using the Relation to Rule Out an Impossible Shape

The relation cuts both ways: given a proposed set of F, V, and E, checking whether they satisfy F+V=E+2 reveals whether such a polyhedron could exist at all. Testing F=10, V=15, E=20:

Since the two sides don't match, no polyhedron with exactly these three counts can exist — not because nobody has found one yet, but because Euler's relation is true for every polyhedron without exception, and any proposed combination that breaks it is thereby ruled out immediately, with no need to attempt drawing it.

A Prism That Looks Like a Cube, But Might Not Be One

Is a square prism the same as a cube? Every cube technically qualifies as a square prism — two square ends, four rectangular sides connecting them. But a square prism's connecting side faces don't have to be squares themselves; they only need to be rectangles. So a square prism is a cube only when those side rectangles happen to be squares too — otherwise it's a taller or shorter square prism that is not a cube. A cube is always a square prism; a square prism is only sometimes a cube.

Why Three Faces Alone Can Never Close Up a Solid

Can a polyhedron have exactly three faces, all of them triangular, and nothing else? No — any solid enclosing a region of space needs at least four faces meeting at no fewer than four vertices, the minimum being the tetrahedron itself (four triangular faces, four vertices, six edges). Three triangles alone can be arranged edge-to-edge, but three flat pieces can never fully enclose a volume on their own — there's always at least one more face required to close the shape into a genuine solid.

Unfolding a Solid Flat, Then Folding It Back Up

A net is what a polyhedron looks like unfolded completely flat — every face laid out separately, still attached along the edges where they'll eventually fold back together. Given several candidate net diagrams, checking which ones actually fold into a valid cube (rather than overlapping faces or leaving gaps) is a genuinely hands-on way to test whether six squares, arranged a particular way on paper, can close up into a solid with no face doubled and no face missing:

Not every arrangement of six squares works — a net has to fold up without any two faces overlapping and without leaving any side of the cube uncovered, and only a specific set of layouts (this cross-like arrangement being one of them) actually manage that.

Quick Recall, One Fact at a Time

A short round of direct questions brings every idea in this lesson together at once:

The cube-versus-cuboid answer reappears twice in this table, and for good reason — it's the same fact from earlier in this lesson (a cube is a regular polyhedron, a cuboid a non-regular one) showing up wherever "same numbers, different shape" is being tested.

The Full Arc of This Chapter

Where Exercise 13.1 practised drawing and counting simple cube stacks, this lesson widened the lens to every polyhedron at once, and Euler's relation is the single fact tying every one of them together, regardless of shape or size. That relationship — connecting a solid's faces, vertices, and edges — is a small preview of how later mensuration chapters connect a solid's dimensions to its surface area and volume, using formulas that likewise hold regardless of a shape's particular size.