1. From 2D to 3D
A plane figure lives on a sheet of paper. It has length and width but no thickness — it has no inside and no outside in three-dimensional space, because three-dimensional space isn't where it lives. A solid changes that. A solid is a region of 3D space, bounded by a surface, with a definite inside.
You can build most of the solids you'll meet by doing one of two things to a 2D shape:
- Translate it — slide a polygon along a straight line off the plane. The polygon sweeps out a prism. A square slides into a cube; a circle slides into a cylinder.
- Revolve it — spin a flat region around a line. The region sweeps out a solid of revolution. A semicircle spun about its diameter becomes a sphere; a right triangle spun about one leg becomes a cone.
A third move — tapering a polygon to a point — gives pyramids and cones. Whichever construction you use, the result is the same kind of object: a closed surface in space with a measurable inside.
Every solid you'll meet in elementary geometry is a polygon that has gotten ambitious. It either grew a height (prism), pinched into an apex (pyramid), or spun into a circle (revolution). The 2D shape never disappears — it becomes the base, and the formulas remember it.
2. The anatomy of a polyhedron
A polyhedron ("many faces") is a solid whose entire surface is made of flat polygons. No curves anywhere — just panels glued edge-to-edge. Three pieces of vocabulary describe every polyhedron.
A face is one of the flat polygonal panels. An edge is a line segment where two faces meet. A vertex (plural vertices) is a point where three or more edges meet.
The letters $V$, $E$, $F$ for the counts of these three things are standard, and they appear in nearly every formula about polyhedra. A cube, for example, has $V = 8$, $E = 12$, $F = 6$.
A polyhedron is convex if the line segment between any two of its points lies entirely inside the solid — no dents, no concavities. All the solids on this page are convex unless flagged otherwise, and most of the classical results (including Euler's formula in its simplest form) apply only to convex polyhedra.
3. Prisms and pyramids
Two families together account for most named polyhedra you'll ever meet.
A polyhedron with two parallel congruent polygonal bases, joined by rectangular (or parallelogram) lateral faces. A prism is named for its base: a triangular prism, a hexagonal prism, and so on. A cube is a square prism whose lateral faces happen to be squares as well.
A polyhedron with one polygonal base and triangular lateral faces meeting at a single point called the apex. A pyramid is also named for its base: a triangular pyramid (also called a tetrahedron), a square pyramid, and so on.
You can read the count of every part directly off the base. If the base is an $n$-gon:
| Faces ($F$) | Vertices ($V$) | Edges ($E$) | |
|---|---|---|---|
| $n$-prism | $n + 2$ (two bases + $n$ sides) | $2n$ (two copies of the base) | $3n$ ($n$ on each base + $n$ vertical) |
| $n$-pyramid | $n + 1$ (one base + $n$ triangles) | $n + 1$ (base vertices + apex) | $2n$ ($n$ on the base + $n$ to the apex) |
Try this on a cube ($n = 4$ prism): $F = 6$, $V = 8$, $E = 12$. Try it on a square pyramid ($n = 4$ pyramid): $F = 5$, $V = 5$, $E = 8$. Plug each into $V - E + F$ and you'll get $2$ both times. That's not a coincidence; we'll get to it in §6.
A prism is called right if its lateral edges are perpendicular to its bases (the sides go straight up). Otherwise it's oblique — the top has been slid sideways without changing shape. The combinatorics ($V, E, F$) are identical; only the geometry differs.
4. Solids of revolution
Replace "polygon" with "any flat region" and "translate or taper" with "spin", and a different family of solids appears. These are the curved solids — they have no flat faces at all (or only circular ones, which don't count as polygons because their boundary isn't a chain of line segments).
- A cylinder is what you get when you slide a circular disc along a straight line — the prism construction applied to a circle. Two circular bases, one curved lateral surface.
- A cone is what you get when you taper a circular disc to a point — the pyramid construction applied to a circle. One circular base, one curved lateral surface, one apex.
- A sphere is what you get when you spin a semicircle about its straight edge. The result has no edges and no vertices at all — just a single closed curved surface.
Other useful members of this family include the frustum (a cone with its tip chopped off, leaving two parallel circular ends of different radii) and the torus (a donut, made by spinning a circle around an external axis). They show up less often in introductory work but are useful to know exist.
For a sphere, the words "face", "edge", and "vertex" simply don't apply — the surface is one continuous curve, with no panels and no creases. Euler's formula, which counts these things, also doesn't apply directly. Solids of revolution sit in a different conceptual bucket from polyhedra.
5. The five Platonic solids
Among all polyhedra, a tiny club has the most symmetry possible. A Platonic solid (or convex regular polyhedron) satisfies three conditions at once:
- It is convex.
- Every face is a congruent regular polygon — the same shape, all sides equal.
- The same number of faces meet at every vertex.
That's a steep set of demands, and remarkably few solids clear all three. There are exactly five — and Euclid proved this at the close of the Elements (Book XIII, around 300 BCE).
Why exactly five?
The argument is short and very pretty. Suppose $p$ regular $q$-gons meet at every vertex of a Platonic solid. Each interior angle of a regular $q$-gon measures $\tfrac{(q-2) \cdot 180^\circ}{q}$. For the vertex to be a corner in 3D — not a flat patch and not a fold that exceeds a full turn — the $p$ angles around it must sum to less than $360^\circ$:
$$ p \cdot \frac{(q-2) \cdot 180^\circ}{q} < 360^\circ $$Rearrange and you get $(p-2)(q-2) < 4$. Since $p \geq 3$ (at least three faces meet at each vertex) and $q \geq 3$ (faces are at least triangles), there are only five integer pairs $(p, q)$ that work:
| $(p, q)$ | Faces meeting · face type | Solid |
|---|---|---|
| $(3, 3)$ | 3 triangles per vertex | Tetrahedron |
| $(4, 3)$ | 4 triangles per vertex | Octahedron |
| $(5, 3)$ | 5 triangles per vertex | Icosahedron |
| $(3, 4)$ | 3 squares per vertex | Cube |
| $(3, 5)$ | 3 pentagons per vertex | Dodecahedron |
The pair $(4, 4)$ — four squares at a vertex — totals exactly $360^\circ$, which flattens the corner into a plane (you get a square tiling, not a solid). Anything past that bulges past $360^\circ$ and can't close up. Five works; six doesn't.
Plato (c. 360 BCE, in the Timaeus) assigned the five solids to the classical elements: tetrahedron / fire, cube / earth, octahedron / air, icosahedron / water, dodecahedron / the cosmos. The physics is wrong but the geometry is exactly right — the five solids really are special, and the fact that ancient Greek mathematicians could prove this without modern algebra is striking.
6. Euler's formula
For every convex polyhedron — whether it's a Platonic solid, an irregular prism, or some weird 17-faced thing you doodled — the counts of vertices, edges, and faces are tied together by a single formula:
$$ V - E + F = 2 $$Try it on every solid we've named so far:
| Solid | $V$ | $E$ | $F$ | $V - E + F$ |
|---|---|---|---|---|
| Tetrahedron | 4 | 6 | 4 | $4 - 6 + 4 = 2$ |
| Cube | 8 | 12 | 6 | $8 - 12 + 6 = 2$ |
| Octahedron | 6 | 12 | 8 | $6 - 12 + 8 = 2$ |
| Dodecahedron | 20 | 30 | 12 | $20 - 30 + 12 = 2$ |
| Icosahedron | 12 | 30 | 20 | $12 - 30 + 20 = 2$ |
| Triangular prism | 6 | 9 | 5 | $6 - 9 + 5 = 2$ |
| Square pyramid | 5 | 8 | 5 | $5 - 8 + 5 = 2$ |
Always $2$. The combinatorial accountancy of vertices, edges, and faces obeys a hidden conservation law.
What it really says
Euler's formula is not really about polyhedra; it's about topology. The number $V - E + F = 2$ depends only on the fact that the polyhedron's surface is topologically a sphere — a closed surface with no holes through it. Stretch a cube into a balloon shape, deform it however you like, and as long as you don't poke a hole through it, the formula still holds for any way of dividing its surface into vertices, edges, and polygons.
The number on the right-hand side, $2$, is the Euler characteristic of the sphere. For a torus (a donut surface, with a hole) it's $0$ instead. So a polyhedron shaped like a picture frame — one with a hole through it — would satisfy $V - E + F = 0$, not $2$. The formula keeps working, but it's tracking the topology, not the geometry.
The statement "$V - E + F = 2$ for any convex polyhedron" is the safe version. The more general truth is "for any polyhedron whose surface is topologically a sphere." Concave (but still sphere-like) polyhedra obey the formula. Polyhedra with holes through them don't.
One use: finding a missing count
If you know any two of $V$, $E$, $F$ for a convex polyhedron, Euler's formula hands you the third. A soccer-ball polyhedron has 12 pentagonal faces and 20 hexagonal ones, so $F = 32$. Each pentagon contributes 5 edges, each hexagon 6 edges, and each edge is shared by exactly two faces, so $E = (12 \cdot 5 + 20 \cdot 6)/2 = 90$. Euler then gives $V = 2 + E - F = 2 + 90 - 32 = 60$ vertices. That number is right — a soccer ball really does have 60 corners, and you didn't have to count them.
7. Nets — unfolding a solid
A net is what you get when you cut a polyhedron along enough edges to lay it flat in a single connected piece, faces still attached at the uncut edges. It's the solid's two-dimensional shadow in the most literal possible way — the same panels, just hinged open.
Nets matter for three reasons:
- They make the surface area concrete. The surface area of the solid is just the total area of its net — which is plane geometry, the kind you already know how to do. (That's the link covered in Surface Area & Volume.)
- They make manufacturing possible. Cardboard boxes, sheet-metal ducts, and origami all rely on unfolding a 3D shape into a flat layout that can be cut from one sheet.
- They help you see the solid. Counting faces on a 3D figure drawn in perspective is error-prone. Counting them on a flat net is trivial.
One solid generally has many nets — different choices of which edges to cut yield different layouts. A cube has 11 distinct nets (counting reflections as the same). A tetrahedron has just 2. Nets are not unique, but the solid they fold back into is.
8. Cross-sections — slicing a solid
A net opens a solid up. A cross-section does the opposite: it slices through the solid with a flat plane and looks at the 2D shape that appears on the cut. Different slices through the same solid produce different cross-sections.
For solids of revolution, cross-sections taken at right angles to the axis of revolution always reproduce the spinning shape:
- Slice a cylinder perpendicular to its axis: every cross-section is a circle (the same circle, in fact).
- Slice a cone perpendicular to its axis: cross-sections are circles whose radius shrinks to zero at the apex.
- Slice a sphere with any plane: every cross-section is a circle. The largest (passing through the center) is a great circle; off-center cuts give smaller circles.
Tilt the slicing plane and the story gets richer. Cutting a cone with a plane that is not perpendicular to its axis produces the conic sections — circle, ellipse, parabola, hyperbola — which connect this elementary topic to the entire theory of quadratic curves and planetary orbits.
If two solids have matching cross-sectional areas at every height, then they have equal volume, even if the solids look completely different. This 17th-century insight (Bonaventura Cavalieri, 1635) prefigures integral calculus and is the cleanest way to derive most volume formulas from scratch.
9. 2D versus 3D measures
Once a solid is on the table, two new numbers come with it. They are easy to confuse and have very different units.
- Surface area is a 2D measure of a 3D object — the total area of the solid's surface. Even though the surface lives in 3D, it is still made of flat or curved sheets, and sheets are two-dimensional. Surface area is measured in square units: $\text{cm}^2$, $\text{m}^2$, and so on.
- Volume is a 3D measure — the amount of space the solid encloses. It is measured in cubic units: $\text{cm}^3$, $\text{m}^3$, and so on.
The units themselves tell you which is which, and they encode something important: surface area scales like length squared, and volume scales like length cubed. Double every dimension of a solid and its surface area goes up by a factor of $4$, while its volume goes up by a factor of $8$. This is the square-cube law, and it explains everything from why ants can lift many times their body weight to why giant animals must be proportioned differently than small ones.
This topic is about what the solids are — their structure, classification, and properties. The companion topic Surface Area & Volume is about how to compute surface area and volume for the common solids (cuboid, cylinder, cone, pyramid, sphere). Pair them.
10. Common pitfalls
A regular prism has a regular polygon as its base (e.g. a regular hexagonal prism). A right prism has its lateral edges perpendicular to the base. They are independent. You can have a regular oblique prism (base is a regular hexagon, but tilted) or a right irregular prism (vertical sides, but the base is a scalene triangle).
A sphere has no flat faces, no edges, and no vertices. Polyhedra are made of flat polygons; spheres aren't. Most theorems about polyhedra — including Euler's formula — don't apply directly to spheres or other solids of revolution.
$V - E + F = 2$ holds for convex polyhedra and, more generally, for any polyhedron whose surface is topologically a sphere. A polyhedron with a tunnel through it (think of a thick picture frame) is topologically a torus, and obeys $V - E + F = 0$ instead. If you plug numbers into Euler's formula and get an answer other than $2$, don't assume the formula is broken — ask whether the solid has a hole through it.
When you count edges from faces, each edge is shared between exactly two faces. So $E = \tfrac{1}{2} \sum (\text{edges per face})$. Forgetting the $\tfrac{1}{2}$ doubles your edge count and gives nonsense answers downstream.
Surface area is in square units; volume is in cubic units. They are not interchangeable, and they never combine. A figure with surface area $24\,\text{cm}^2$ and volume $8\,\text{cm}^3$ has those numbers honestly — they are not the "same" $24$ and $8$, because they're measuring different things.
11. Worked examples
Example 1 · Count $V$, $E$, $F$ for a hexagonal prism, then verify Euler's formula
A hexagonal prism has two hexagonal bases joined by six rectangular faces. Use the prism counts from §3 with $n = 6$:
- Faces: $F = n + 2 = 8$.
- Vertices: $V = 2n = 12$.
- Edges: $E = 3n = 18$.
Check Euler:
$$ V - E + F = 12 - 18 + 8 = 2 \;\checkmark $$Example 2 · A polyhedron has 20 vertices and 30 edges. How many faces?
Solve Euler's formula for $F$:
$$ F = 2 - V + E = 2 - 20 + 30 = 12. $$Twelve faces. (This is a dodecahedron, if all the faces are congruent regular pentagons — but the formula didn't need to know that.)
Example 3 · Why isn't there a Platonic solid made of regular hexagons?
A regular hexagon has interior angle $120^\circ$. To form a corner of a 3D solid at a vertex, the angles around that vertex must sum to less than $360^\circ$. With three hexagons:
$$ 3 \cdot 120^\circ = 360^\circ $$That's exactly a full turn, which lies flat (it tiles the plane — think of a honeycomb). Two hexagons isn't enough to form a corner. So no Platonic solid can have hexagonal faces, and the same argument rules out anything with seven or more sides per face.
Example 4 · How many distinct cross-sections can a cube produce?
It depends on the angle of the cut. The possibilities are:
- Triangle — slice a corner off (the plane cuts three adjacent edges).
- Quadrilateral — most "generic" cuts; can be a square, rectangle, rhombus, or trapezoid depending on orientation.
- Pentagon — a plane that cuts five of the six faces.
- Hexagon — a plane perpendicular to a long diagonal and through the center cuts all six faces, producing a regular hexagon.
So a cube admits cross-sections with 3, 4, 5, or 6 sides — but never 7 or more, because a cube only has 6 faces, and each face can contribute at most one edge to the cross-section.
Example 5 · A net is six squares in a row. Can it fold into a cube?
No. A cube needs three pairs of opposite faces — top/bottom, front/back, left/right — and a straight line of six squares folds into an open spiral, not a closed box. Of the $11$ valid cube nets, the most common is the "cross" (a column of four squares with one square attached to each side of the second square). Six in a row is one of the most common invalid guesses.
Test for yourself: which two opposite faces would the second-from-left and second-from-right squares fold to? You'll find they end up adjacent, not opposite.