Polyhedra
Some solids are built only from flat pieces and some curve. What makes the flat-sided ones a family of their own, and is there a rule hiding in how many faces, edges and corners they have?
Where you are 7 in Three-dimensional shapes
Why this exists
Put a sugar cube, a soup can and a marble on a table. The sugar cube is made only of flat pieces with straight creases between them. The can has two flat ends, but its side curves. The marble curves everywhere. A solid of the first kind can be described completely by flat Polygons: you can count its pieces, name their shapes, cut them from card and fold them up. A solid with a curve cannot be taken apart into flat pieces that way.
Flat-sided solids turn up wherever nature or people build with straight cuts. A grain of table salt grows as a tiny cube. A quartz crystal grows as a six-sided column with a pointed tip. A jeweler cuts a gemstone into many flat, shiny faces, and a geodesic dome is a round-looking roof made of flat triangles. This lesson names that family and counts its parts. The counting turns up a surprise: for every one of these solids without a hole through it, the numbers of faces, edges and corners obey one simple rule.
Definition
A polyhedron is a closed solid whose whole surface is made of flat polygons. The plural is polyhedra, and the name is Greek for roughly “many faces”. Its parts have names (the earlier lessons Faces, edges and vertices and Naming and describing solid shapes meet them on everyday solids):
- A face is one of the flat polygons that make up the surface.
- An edge is a straight segment where two faces meet.
- A vertex (plural vertices) is a corner point where edges meet.
We write F for the number of faces, V for the number of vertices and E for the number of edges.
Two words in the definition do real work. Closed means the surface has no gaps: the solid seals its inside in, like a box with its lid on. Whole means every part of the surface is flat. A solid with any curved part is not a polyhedron, however many flat faces it also has. A cylinder has two flat circles but a curved side, a cone has a flat circle and a curved side (both are studied in Cylinders and cones), and a sphere curves all over. None of the three is a polyhedron. A circle is not a polygon either, since it has no straight sides.
Intuition
Figure 1 shows the three parts on a cube. Run a finger along an edge of a real box: there is one face on each side of your finger and never a third. So every edge is the meeting line of exactly two faces. At a vertex, three or more faces come together.
Counting the parts of a solid is easy to get wrong by skipping one or counting one twice. A steady way is to count in layers. Stand the solid on a face, count what lies on the bottom, then what lies on the top, then what runs between them, and say each layer out loud.
Figure 2 sorts four solids. The first two have only flat faces, so they are polyhedra. The cylinder and the cone each have a curved surface, so they are not, even though each has a flat circle.
Results
Edges from faces. Add up the numbers of sides of all the faces. Each edge is a side of exactly two faces, so this total counts every edge twice:
Euler’s formula. For every polyhedron without a hole through it,
The rule is named after the eighteenth-century mathematician Leonhard Euler. It works as a check on a count, and it also finds a missing count: if two of F, V and E are known, the third is the number that makes F + V exactly 2 more than E.
Why it is true
The edge rule needs only the fact from the Intuition: each edge borders two faces, so when every face reports its sides, each edge is reported twice.
A full proof of Euler’s formula is longer; it goes through flat drawings of dots and lines, the subject of Planar graphs and Euler's formula. Here is why the total does not budge when a solid is changed. A cube has 6 + 8 - 12 = 2. Slice off one corner of the cube, where three faces meet. The cut makes one new triangle face, so F goes up by 1. The old corner is gone, but the cut crosses three edges and makes three new corners, so V goes up by 3 - 1 = 2. The triangle brings three new edges, so E goes up by 3. Altogether F + V - E changes by 1 + 2 - 3 = 0.
Gluing a pyramid onto a square face behaves the same way. The square disappears inside and four triangles appear (F up by 3), the tip is new (V up by 1), and four slanting edges are new (E up by 4): the change is 3 + 1 - 4 = 0. Moves like these never change the total, and that idea is the heart of the full proof.
Worked examples
A salt crystal
Zdenek looks at grains of table salt under a magnifying glass, and each one is a tiny cube. Find F, V and E for a cube, and check Euler’s formula.
SolutionStand the cube on one face and count in layers. Faces: one on the bottom, one on top and four round the sides, so F = 1 + 1 + 4 = 6. Vertices: four round the bottom and four round the top, so V = 4 + 4 = 8. Edges: four round the bottom square, four round the top square and four upright edges joining them, so E = 4 + 4 + 4 = 12.
Check. F + V - E = 6 + 8 - 12 = 2. The edge rule agrees too: six squares have 6 \times 4 = 24 sides, and 24 \div 2 = 12.
Polyhedron or not
Ahanu sorts a tray at a rock shop. Which of these are polyhedra? (a) A quartz crystal: a six-sided column with a flat base and a point made of six triangles. (b) A cabochon: a gem polished into a smooth rounded top over a flat oval base. (c) A pearl. (d) A diamond crystal made of eight flat triangles, shaped like two square pyramids joined at their bases.
SolutionThe only question is whether every part of the surface is flat; size, shine and color do not matter. (a) Yes: the base, the six sides and the six triangles are all flat polygons and together they close the solid. (b) No: the flat base does not help, because the top curves. (c) No: a pearl is close to a sphere and curves everywhere. (d) Yes: eight flat triangles close it up.
Counting a quartz crystal
Figure 3 shows a quartz crystal: a six-sided column with a hexagon at the bottom, topped by a point made of six triangles. Find F, V and E.
Count in layers, bottom to top. Faces: the hexagon at the bottom, six rectangles round the column and six triangles in the point, so F = 1 + 6 + 6 = 13. Vertices: six round the bottom, six where the column meets the point, and the tip, so V = 6 + 6 + 1 = 13. Edges: six round the bottom, six running up the column, six round the top of the column and six slanting up to the tip, so E = 6 + 6 + 6 + 6 = 24.
Check. F + V - E = 13 + 13 - 24 = 2. By the edge rule, the faces have 6 + 6 \times 4 + 6 \times 3 = 48 sides, and 48 \div 2 = 24.
Edges from the faces of a cut gem
Blessing cuts a gem shaped like two five-sided pyramids joined base to base (Figure 4). It has ten triangle faces, five round the top point and five round the bottom point. Find E without tracing the edges one by one, then find V and check Euler’s formula.
Ten triangles have 10 \times 3 = 30 sides between them. Every edge is a side of two of the triangles, so
The vertices are the five corners round the middle and the two points, so V = 5 + 2 = 7.
Check. F + V - E = 10 + 7 - 15 = 2.
A missing count from Euler’s formula
Gustavo has a mineral crystal whose surface is six squares and eight triangles (Figure 5). Find E, then use Euler’s formula to find V. Then check V another way: at every corner, two squares and two triangles meet.
The faces have 6 \times 4 + 8 \times 3 = 24 + 24 = 48 sides, so E = 48 \div 2 = 24. There are F = 6 + 8 = 14 faces.
Euler’s formula says F + V is 2 more than E, so F + V = 24 + 2 = 26. With F = 14,
Check. The faces have 48 corners between them, and each vertex of the crystal is a corner of four faces, so it was counted four times: 48 \div 4 = 12. Both ways give V = 12.
Struts and hubs of a geodesic dome
Hanneke is building a geodesic dome as a garden climbing frame (Figure 6). Its surface is 40 flat triangle panels, and it stands on a floor whose outline is a ten-sided polygon. Each edge is a metal strut and each vertex is a hub where struts bolt together. How many struts and hubs does she need?
The dome on its own is open at the bottom, so it is not closed. Count the floor as one more face and it becomes a closed polyhedron with F = 40 + 1 = 41. The floor adds no struts or hubs: its ten sides are the struts lying on the ground, and its ten corners are the hubs on the ground.
The faces have 40 \times 3 + 10 = 130 sides, so E = 130 \div 2 = 65. By Euler’s formula, F + V = 65 + 2 = 67, so
Hanneke needs 65 struts and 26 hubs. The dome looks rounded, but because every panel is flat it is a polyhedron, not half of a sphere.
Boundaries
A curved surface rules a solid out. A cylinder, cone or sphere is not a polyhedron, and Euler’s formula says nothing about it. Calling a cylinder’s two circles and curved side “faces” and its two rims “edges” gives 3 + 0 - 2 = 1, not 2.
The surface must be closed. A box with no lid has 5 faces, 8 vertices and 12 edges, and 5 + 8 - 12 = 1. That is why the dome in Example 6 needed its floor counted as a face.
A hole through the solid breaks the formula. For a polyhedron shaped like a square picture frame, F + V - E = 0.
Where this goes next
The most common polyhedra come in two families, built and named in Prisms and pyramids: prisms, with two matching ends joined by flat sides, and pyramids, with one base and a point. The five regular solids asks which polyhedra can have every face the same regular polygon, and finds only five. Because every face is flat, a polyhedron can be cut open and laid out as a flat pattern of polygons, which is the idea behind Nets of solids and Surface area of prisms and cylinders. Slicing a polyhedron always leaves a polygon, the subject of Cross sections of solids. And Euler’s formula returns for networks of dots and lines drawn flat on paper in Planar graphs and Euler's formula.
Exercises
Polyhedron or not
- Which of these are polyhedra? Give a reason for each. (a) a sugar cube (b) a soup can (c) a closed tent with two triangle ends, two rectangle sides and a rectangle floor (d) an ice-cream cone (e) a marble
- A solid has five flat faces and one curved face. Is it a polyhedron? Explain.
Counting
For Exercises 3 to 8, find F, V and E, then check Euler’s formula.
- A pyramid whose four faces are all triangles.
- A pyramid with a square base and four triangle faces.
- A crystal shaped as a three-sided prism: two triangle ends joined by three rectangles.
- A crystal shaped as a six-sided column with a flat hexagon at each end.
- The crystal in Figure 7: a square column with a four-sided pyramid point at each end.
- A pyramid with a pentagon base and five triangle faces.
Edges from faces
- A cut stone has 6 square faces and 8 hexagon faces. Find E, then use Euler’s formula to find V.
- A cut stone has 2 octagon faces and 16 triangle faces. Find E and V.
- Can a polyhedron have exactly 7 faces, all of them triangles? Explain.
Using Euler’s formula
- A polyhedron has F = 10 and E = 24. Find V.
- A polyhedron has V = 18 and E = 27. Find F.
- A polyhedron has F = 20 and V = 36. Find E.
- A gem catalog lists a stone with F = 9, V = 14 and E = 20. Explain why at least one of the three counts must be wrong.
- A small garden dome is made of 15 triangle panels standing on a five-sided floor. How many struts (edges) and hubs (vertices) does it have?
Reasoning
- Explain why adding up the sides of all the faces counts every edge exactly twice.
- A glass paperweight has a flat round bottom and a smooth curved top. Explain why it is not a polyhedron.
- Explain why every polyhedron has at least four faces.
- A three-sided prism has F = 5, V = 6 and E = 9. A pointed cap with three triangle sides is glued onto one triangle end, so that end is hidden inside. Without counting the new solid from scratch, find its F, V and E, and check Euler’s formula.
Answers
- (a) and (c) are polyhedra: every face is a flat polygon and the surface is closed. (b) is not, because the can’s side curves; (d) is not, because a cone’s side curves; (e) is not, because a marble curves all over.
- No. A polyhedron’s whole surface must be flat polygons, and one curved face is enough to rule it out.
- F = 4, V = 4, E = 6: three edges round the bottom and three rising to the tip. 4 + 4 - 6 = 2.
- F = 5, V = 5, E = 8: four edges round the base and four rising to the tip. 5 + 5 - 8 = 2.
- F = 5, V = 6, E = 9: three edges round each end and three joining the ends. 5 + 6 - 9 = 2.
- F = 8, V = 12, E = 18: two ends and six sides; six corners on each end; six edges round each end and six joining them. 8 + 12 - 18 = 2.
- F = 12, V = 10, E = 20: four rectangles and four triangles at each end; eight corners on the column and two points; eight edges round the two ends of the column, four up its sides and eight slanting to the points. 12 + 10 - 20 = 2.
- F = 6, V = 6, E = 10: five edges round the base and five rising to the tip. 6 + 6 - 10 = 2.
- E = 36, V = 24. The faces have 6 \times 4 + 8 \times 6 = 72 sides, so E = 72 \div 2 = 36. With F = 14, F + V = 36 + 2 = 38, so V = 38 - 14 = 24.
- E = 32, V = 16. The faces have 2 \times 8 + 16 \times 3 = 64 sides, so E = 32. With F = 18, F + V = 34, so V = 34 - 18 = 16.
- No. Seven triangles have 7 \times 3 = 21 sides. Each edge accounts for exactly two sides, so the total must be an even number, and 21 is odd.
- V = 16. F + V = 24 + 2 = 26, so V = 26 - 10 = 16.
- F = 11. F + V = 27 + 2 = 29, so F = 29 - 18 = 11.
- E = 54. E is 2 less than F + V = 56, so E = 56 - 2 = 54.
- 9 + 14 - 20 = 3, not 2. A cut stone has no hole through it, so its counts must give 2. (If F and V are right, the stone has E = 21.)
- 25 struts and 11 hubs. Counting the floor, F = 16; the faces have 15 \times 3 + 5 = 50 sides, so E = 25; then F + V = 27, so V = 27 - 16 = 11.
- Each edge is the line where exactly two faces meet, so it is a side of each of those two faces. When every face reports its sides, the edge is reported once by each, twice in all.
- Its top is curved, and a polyhedron’s whole surface must be made of flat polygons. Its round bottom is not a polygon either.
- At any vertex at least three faces meet, since two flat faces alone meet only along one fold and cannot make a corner. Three faces round one corner spread out like the corner of a box and leave the far side open, so at least one more face is needed to close the solid.
- F = 7, V = 7, E = 12. The hidden end is lost and three triangles are added, so F = 5 - 1 + 3 = 7; the cap’s point is the only new corner, so V = 6 + 1 = 7; its three slanting edges are new, so E = 9 + 3 = 12. 7 + 7 - 12 = 2.