A dice, a brick, a pyramid in Egypt, a football, a crystal of salt: all of them are solids bounded by flat polygons. Such a solid is a polyhedron, and the first questions about one are counting questions. How many faces, how many edges, how many corners? This chapter counts them without drawing every piece: with formulas for prisms and pyramids, with the trick of counting each edge twice, and with a relation Euler found in 1750 that ties the three numbers together. It ends with the other everyday skill about solids: reading and drawing them from above, from the front and from the side.
By the end you can
- Name the faces, edges and vertices of a polyhedron and count them.
- Give V, E and F of any prism or pyramid from the number of sides of its base.
- Find the number of edges by adding the sides of the faces, or the edges at each vertex.
- Use Euler's relation V − E + F = 2 to find a missing count.
- Draw and read the top, front and side views of a solid.
A polyhedron (Greek for "many seats") is a solid whose surface is made of flat polygons. Each polygon is a face. Two faces meet along a segment, an edge, and every edge belongs to exactly two faces. The points where edges meet are the vertices, the corners. The counts are written V (vertices), E (edges) and F (faces). A cube has 8 vertices, 12 edges and 6 faces. A polyhedron is convex when it has no dents: the segment between any two of its points stays inside it. A dice and a pyramid are convex; a star-shaped solid is not.
Each face lists its sides, and every edge is a side of exactly two faces, so adding up the sides of all the faces counts every edge twice. The same happens with the vertices: each edge has two ends. Cut the corners and watch the counts: every corner becomes a new face, every edge keeps a shorter middle piece, and V − E + F stays 2. The icosahedron cut a third of the way along its edges is the classic football. Drag to turn it; hidden edges are dashed.Swipe to turn it; hidden edges are dashed.
In the figure, switch between faces, edges and vertices to see each set on its own, and turn the solid: the dashed lines are edges behind it. Counting by eye works for a cube and fails quickly for a dodecahedron. The rest of the chapter replaces counting with reasoning.
A prism has two equal polygons, its bases, joined by a band of parallelograms (rectangles when it stands upright). A pyramid has one base and a single top vertex, the apex, joined to every corner of the base by triangles. Both are named after the base: a pentagonal prism has pentagons as bases. If the base has n sides, every count follows from n. For the prism: n vertices on each base, so 2n; n edges around each base plus n edges going up, so 3n; n side faces plus 2 bases, so n + 2. For the pyramid: n base vertices plus the apex; n base edges plus n edges up to the apex; n triangles plus the base.
| Solid | n | V | E | F |
|---|---|---|---|---|
| triangular prism | 3 | 6 | 9 | 5 |
| cube (square prism) | 4 | 8 | 12 | 6 |
| hexagonal prism | 6 | 12 | 18 | 8 |
| triangular pyramid | 3 | 4 | 6 | 4 |
| square pyramid | 4 | 5 | 8 | 5 |
| hexagonal pyramid | 6 | 7 | 12 | 7 |
Most solids in exercises are neither prisms nor pyramids, and the question gives the faces instead: "a solid has 12 pentagonal faces and 20 hexagonal faces". Go round each face and count its sides. Every side is an edge, but every edge is a side of two faces, so this count meets each edge exactly twice. The same double count works from the corners: at each vertex count the edges that meet there (the degree of the vertex); every edge has two ends, so the total is again 2E.
Add the vertices, take away the edges, add the faces. For the cube, 8 − 12 + 6 = 2; for the football, 60 − 90 + 32 = 2; for every polyhedron in the figure, 2. Leonhard Euler noticed this in 1750, and it holds for every convex polyhedron. The idea of the proof: look at the solid through one face, very close, and its edges become a flat drawing with no crossings, in which the face you look through is the outside. Building that drawing one edge at a time never changes V − E + F. The full proof is in the Planar Graphs chapter of the Discrete Mathematics section; here the relation is a tool.
Finding the missing count
The standard exercise gives the faces and asks for the vertices. Two steps: the faces give E by the double count, then Euler gives V. A convex polyhedron has 8 triangular faces and 6 square faces. How many vertices?
The solid is real: it is the cube with its corners cut off halfway along each edge (the cuboctahedron), and at each of its 12 vertices 4 edges meet, so 4 · 12 = 48 = 2E. A second count that agrees is the best check.
Saw off every corner of a polyhedron, a little way along each edge. Each old corner leaves a new small face, a polygon with as many sides as there were edges meeting there. Each old edge loses both its ends but keeps its middle piece, and gets one new vertex at each end. Press "cut the corners" in the figure and follow the counts. The new numbers, marked with a prime, come straight from the old ones.
Cut the 12 corners of an icosahedron (20 triangles, 5 meeting at each corner) a third of the way along each edge: the triangles become hexagons, the corners become 12 pentagons. V' = 2 · 30 = 60, E' = 3 · 30 = 90, F' = 20 + 12 = 32. That is the football of the earlier example, and its official name is the truncated icosahedron.
A polyhedron is regular when all its faces are the same regular polygon and the same number of faces meet at every vertex. There are only five, known as the Platonic solids. The faces at one corner must add up to less than 360°, or they would lie flat: three, four or five equilateral triangles (60° each), three squares (90°) or three regular pentagons (108°) are the only options. The Planar Graphs chapter proves the same thing from Euler's relation.
| Solid | Face | Faces at each vertex | V | E | F |
|---|---|---|---|---|---|
| tetrahedron | triangle | 3 | 4 | 6 | 4 |
| cube (hexahedron) | square | 3 | 8 | 12 | 6 |
| octahedron | triangle | 4 | 6 | 12 | 8 |
| dodecahedron | pentagon | 3 | 20 | 30 | 12 |
| icosahedron | triangle | 5 | 12 | 30 | 20 |
Look at the columns: the cube and the octahedron swap V and F, and so do the dodecahedron and the icosahedron. Put a point at the centre of every face of a cube and join neighbouring points: you get an octahedron. Such pairs are called dual. The name of each solid counts its faces: tetra 4, hexa 6, octa 8, dodeca 12, icosa 20.
Plans for a building or a machine part do not draw the solid in perspective. They draw what you see looking straight at it from three directions: from above (the top view, or plan), from the front, and from one side. Each view is an orthogonal projection: every point of the solid moves straight towards a flat wall, along lines perpendicular to it, and lands where it hits. Depth is lost, so a view shows only the outline in that direction, and whatever stands behind something taller disappears.
A view keeps only what reaches the wall: in the front view each column shows its tallest stack, wherever it stands in depth, and in the side view each row does the same. Different solids can share a view, which is why plans give two or three of them. The top view also builds the solid: tap a square to stack one more cube; the numbers are the heights. Drag the solid to turn it.Swipe the solid to turn it.
In the figure, each column of the front view shows the tallest stack in that column, wherever it stands; the side view does the same for each row. Try to build two different solids with the same front view: it is easy, which is why one view is never enough to describe a solid. Seen from the right, the front of the solid is on the left of the picture.
| Solid | Top view | Front view | Side view |
|---|---|---|---|
| upright cylinder | circle | rectangle | rectangle |
| upright cone | circle with its centre marked (the apex) | triangle | triangle |
| sphere | circle | circle | circle |
| square pyramid | square with its two diagonals (the slanted edges) | triangle | triangle |
| triangular prism lying on a side (a tent) | rectangle split by a line (the ridge) | triangle | rectangle |
Projecting onto the coordinate planes
With coordinates the views become very simple. Put the solid in space with x to the right, y up and z towards you. Projecting onto a coordinate plane just forgets one coordinate: the top view keeps x and z, the front view keeps x and y, the side view keeps y and z. A segment parallel to the wall keeps its length; a tilted one comes out shorter, multiplied by the cosine of its angle with the wall; one pointing straight at the wall shrinks to a point. In the same way a circle tilted towards you looks like an ellipse, and seen edge-on it is a segment.
1. A prism has 24 edges. What is its base, and how many faces does it have? E = 3n = 24 gives n = 8: an octagonal prism, with F = n + 2 = 10 faces and V = 16 vertices. Check: 16 − 24 + 10 = 2.
2. A convex polyhedron has 20 vertices, and 3 edges meet at each one. How many faces? 2E = 3 · 20 = 60, so E = 30, and F = 2 − V + E = 2 − 20 + 30 = 12. If all the faces are alike, each has 2E/F = 60/12 = 5 sides: the dodecahedron.
3. A gem is cut with 10 triangular faces and 2 pentagonal faces. Edges: 2E = 10 · 3 + 2 · 5 = 40, E = 20. Vertices: V = 2 + 20 − 12 = 10. (It is a pentagonal antiprism: two pentagons, one turned, joined by a zigzag band of triangles.)
4. A pyramid has 10 faces. Then n + 1 = 10, the base is a 9-sided polygon, E = 2 · 9 = 18 and V = 10.
5. A cube has every corner cut off. V' = 2 · 12 = 24, E' = 3 · 12 = 36, F' = 6 + 8 = 14 (6 octagons and 8 triangles). 24 − 36 + 14 = 2.
6. Can a polyhedron have exactly 7 edges? With F faces of at least 3 sides, 2E ≥ 3F, so F ≤ 4; with corners of at least 3 edges, V ≤ 4 as well. Then V − E + F ≤ 4 − 7 + 4 = 1, never 2. No polyhedron has 7 edges, although 6 (tetrahedron) and 8 (square pyramid) both exist.
| Wrong | Right | Why |
|---|---|---|
| football: E = 12 · 5 + 20 · 6 = 180 | E = 180/2 = 90 | every edge is a side of two faces, so the sides count it twice |
| V − E + F = 2 for a frame with a hole | V − E + F = 0 | the relation is for solids without tunnels, such as every convex one |
| a hexagonal prism has F = 6 | F = 6 + 2 = 8 | the two bases are faces too |
| a pyramid with n-sided base has 2n vertices | n + 1 | only one point, the apex, is added to the base |
| front view of a back stack hidden by a lower front one | the taller stack shows | a view keeps the tallest thing in each line of sight, front or back |
Key ideas
- Prism with n-sided base: V = 2n, E = 3n, F = n + 2. Pyramid: V = n + 1, E = 2n, F = n + 1.
- Adding the sides of all the faces counts every edge twice: 2E = 3F₃ + 4F₄ + …; the same from the vertices.
- Euler: V − E + F = 2 for every convex polyhedron; two counts give the third.
- Cutting every corner: V' = 2E, E' = 3E, F' = F + V.
- There are five regular solids: tetrahedron, cube, octahedron, dodecahedron, icosahedron.
- A view is an orthogonal projection: it drops the depth and keeps the tallest thing in each line of sight.