Volumes and surface areas of regular polyhedra

A polyhedron is a solid whose boundary is made of plane figures — polygons, not curved walls. The prism, cylinder, and ball sit in the previous table; a cylinder has a curved side, a ball has no faces.

A regular polyhedron (a Platonic solid) adds two extra constraints: every face is a regular $n$-gon of the same side $a$, and at every vertex the same number of faces meet. Drop either and the solid leaves this list: a rectangular box has rectangular faces, but the three edges need not be equal; a square pyramid mixes a square with triangles.

Those two constraints leave five solids, not five unrelated topics. The common edge $a$ is the only length; $V$ scales as $a^3$ and $O$ as $a^2$. Drag $B$. $A$ stays put, so $AB$ is that edge. Turn the figures.

Euler's formula

Count vertices $c$, edges $e$, and faces $f$. For a regular polyhedron the table records

$c-e+f=2.$

The same identity holds on every row. A tetrahedron has $4-6+4=2$. A cube has $8-12+6=2$. Walk the rest — the last column never leaves $2$.

Two pairings sit in the counts. The cube has $(c,f)=(8,6)$ and the octahedron has $(6,8)$: vertices and faces swap, the edge count stays $12$. The dodecahedron and the icosahedron swap the same way, with $e=30$ on both. The tetrahedron matches itself: $c=f=4$. The formula is a relation among three integers, not a construction. The boards below still only drag the edge $a$; each board freezes which regular $n$-gon meets how many times at a vertex.

Three triangles at a vertex

Four equilateral triangles. Three of them meet at each vertex. The surface is four copies of the equilateral area $\frac{\sqrt{3}}{4}{a}^2$:

$V=\frac{\sqrt{2}}{12}{a}^3,O=\sqrt{3}{a}^2.$

Among the five this is the fewest faces. Drag $B$; the other two vertices follow so every edge stays $a$.

Three squares at a vertex

Six squares, every edge $a$. The same solid as in the volume table, now read as a Platonic solid: three squares at a vertex, eight vertices. Each face has area $a^2$, so the skin is six of them:

$V=a^3,O=6a^2.$

$V=a^3$ is the volume of a box whose three edges have collapsed to one length. Drag $B$; the other six vertices follow.

Four triangles at a vertex

Eight equilateral triangles — twice as many as the tetrahedron, the same triangular tile. Four triangles meet at a vertex. The surface is eight copies of $\frac{\sqrt{3}}{4}{a}^2$, which is twice the tetrahedron's skin:

$V=\frac{\sqrt{2}}{3}{a}^3,O=2\sqrt{3}{a}^2.$

The volume is not eight tetrahedron volumes ($8\times \sqrt{2}/12=2\sqrt{2}/3$). The two solids share a face shape, not a dissection. In the Euler table this is the cube with $c$ and $f$ swapped. Drag $B$; the remaining vertices follow so every edge stays $a$.

Three pentagons at a vertex

Twelve regular pentagons. Three pentagons meet at a vertex. This is the only one of the five whose faces are not triangles or squares.

$V=7.663a^3,O=20.646a^2.$

Live $V$ is the exact volume; it rounds to $7.663a^3$. The surface on the board is the closed form $3\sqrt{25+10\sqrt{5}}{a}^2$, which rounds to $20.646a^2$. Drag $B$ — twelve pentagons is a busy skin, so only $A$ and $B$ keep labels.

Five triangles at a vertex

Twenty equilateral triangles. Five triangles meet at a vertex. The surface is twenty copies of $\frac{\sqrt{3}}{4}{a}^2$, five times the tetrahedron's skin:

$V=2.182a^3,O=8.660a^2.$

Those two numbers are again the table's approximations. Live $V$ is exact and rounds to $2.182a^3$. The surface on the board is $5\sqrt{3}{a}^2$, which rounds to $8.660a^2$. In the Euler table this is the dodecahedron with $c$ and $f$ swapped.

The icosahedron's symmetry is the subject of a famous book by Felix Klein: he tied that solid to the quintic equation. This page keeps the volume and the surface; it does not open that book.