Faces, edges and vertices
A polyhedron is a solid whose faces are all flat polygons. A cube, a cuboid, a prism and a pyramid are polyhedra; a cylinder, a cone and a sphere are not, because each has a curved surface.
Every polyhedron has three kinds of part. A face is one of its flat sides. An edge is a straight line where two faces meet. A vertex is a corner, a point where edges meet; the plural is vertices.
A cube has 8 vertices: 4 around its top face and 4 around its bottom face. In the drawing seven can be seen; the eighth is at the back, hidden behind the cube.
The cube’s 8 vertices, numbered so they can be counted.
Counting the edges and the faces
The cube has 12 edges: 4 around the top, 4 around the bottom and 4 standing upright between them. Three of the 12 meet at the hidden vertex and run behind the cube, so they are drawn dashed.
It has 6 faces: top and bottom, front and back, left and right. Three face toward you, and three face away.
Now combine the three counts. Take the number of vertices, subtract the number of edges and add the number of faces: 8 − 12 + 6 = 2.
The cube’s 12 edges. The three drawn dashed run behind it.
The cube’s 6 faces, three toward you and three away. So 8 − 12 + 6 = 2.
Lengths do not matter
Stretch the cube into a cuboid. Its edges change length, but it still has 8 vertices, 12 edges and 6 faces, joined in the same way. So 8 − 12 + 6 = 2 again.
The counts depend only on how the faces, edges and vertices are joined together, not on how long the edges are or what angles the faces make.
Stretched into a cuboid, the solid still has 8 vertices.
A different solid gives 2 as well
A square pyramid is joined together differently. It has 5 vertices: 4 around the square base and 1 at the apex. It has 8 edges: 4 around the base and 4 rising to the apex. It has 5 faces: the square and 4 triangles. And 5 − 8 + 5 = 2.
An octahedron is made of 8 equilateral triangles, like two square pyramids joined base to base. It has 6 vertices: 4 around the middle, 1 at the top and 1 at the bottom. It has 12 edges: 4 around the middle, 4 rising to the top and 4 falling to the bottom. It has 4 faces above the middle and 4 below, and 6 − 12 + 8 = 2.
The same answer comes out every time. This is Euler’s formula, described by the Swiss mathematician Leonhard Euler in 1750: for any polyhedron with no holes through it, V − E + F = 2, where V is the number of vertices, E the number of edges and F the number of faces.
The square pyramid’s 8 edges: 4 around the base and 4 rising to the apex. It has 5 vertices and 5 faces, and 5 − 8 + 5 = 2.
Why the answer does not change
Here is a reason to expect the same answer. Change a polyhedron into a new one, and watch what happens to V − E + F.
Slice one corner off the cube. The corner vertex is gone, but the cut makes a new triangular face with 3 new vertices and 3 new edges. So V goes up by 3 − 1 = 2, E goes up by 3 and F goes up by 1, and V − E + F changes by 2 − 3 + 1 = 0. The new solid has 10 vertices, 15 edges and 7 faces, and 10 − 15 + 7 = 2.
Stand a square pyramid on the top face of the cube, like the roof of a house. That adds 1 vertex at the top and 4 edges rising to it. The square top face is covered, but 4 triangles take its place, so F goes up by 4 − 1 = 3. The change is 1 − 4 + 3 = 0: the house has 9 vertices, 16 edges and 9 faces, and 9 − 16 + 9 = 2.
The same happens with any corner and any face. Slicing off a corner where n edges meet adds n − 1 vertices, n edges and 1 face, and (n − 1) − n + 1 = 0. A pyramid on a face with n sides adds 1 vertex, n edges and n − 1 faces, and 1 − n + (n − 1) = 0. Changes like these build many polyhedra from a cube, and none of them can move the 2. Showing that every polyhedron without holes can be built this way is much harder, which is why this is a reason to believe the formula and not yet a proof of it.
When the formula fails
The formula needs a solid with no hole through it. A square picture frame has one. Build it from a thick square block with a square hole cut through the middle, and split its top and its bottom into 4 flat faces each, since a face must be a polygon and a square with a hole in it is not one. It has 8 vertices on the outside and 8 around the hole, which is 16. It has 16 faces: 4 on top, 4 underneath, 4 around the outside and 4 around the hole. And it has 32 edges: 12 on the top face, 12 on the bottom and 8 upright. So V − E + F = 16 − 32 + 16 = 0, not 2.
Using the formula
Knowing two of the three counts gives the third. A polyhedron with 12 vertices and 18 edges has F = 2 − V + E = 2 − 12 + 18 = 8 faces.
When the solid is too large to count by eye, count the edges face by face and halve: every edge belongs to exactly two faces, so counting the edges of every face counts each edge twice.
Worked example: A Soccer Ball Stitched from Pentagons and Hexagons: Its Seams and Its Corners
Question A soccer ball is stitched together from 32 leather panels: 12 regular pentagons and 20 regular hexagons. Each seam joins an edge of one panel to an edge of another, and the seams meet at corners. (a) How many seams are there? (b) Use Euler's formula V − E + F = 2 to find the number of corners. The same number of panels meet at every corner: how many?
1.Count the edges of the panels, one kind at a time. The 12 pentagons have 12 × 5 = 60 edges and the 20 hexagons have 20 × 6 = 120 edges, which is 60 + 120 = 180 panel edges in all.
One pentagon and the five hexagons round it, laid flat. All the panels have 12 × 5 + 20 × 6 = 180 edges. 2.(a) Each seam joins two panel edges, so counting panel edges counts every seam twice. There are 180 ÷ 2 = 90 seams.
(a) Each seam joins two panel edges, like each gold edge here, so there are 180 ÷ 2 = 90 seams. 3.The panels are the faces, so F = 12 + 20 = 32, and the seams are the edges, so E = 90. Euler's formula gives V − 90 + 32 = 2.
The panels are the faces and the seams the edges: V − 90 + 32 = 2. 4.So V = 2 + 90 − 32 = 60. The ball has 60 corners.
So the ball has V = 60 corners. 5.(b) The panels have 12 × 5 + 20 × 6 = 180 corners between them, and these gather at the 60 corners of the ball. So 180 ÷ 60 = 3 panels meet at each corner. The ball has 60 corners, with 3 panels at each.
(b) 180 panel corners gather at 60 corners, 3 at each. Laid flat they leave a gap, which closes when the ball is stitched.
Answer: (a) 90 seams; (b) 60 corners, with 3 panels meeting at each
Common mistakes
- Taking 180 as the number of seams. Every seam runs along the edges of two panels, so the count of panel edges is twice the number of seams.
- Putting F = 12 or F = 20, the number of one kind of panel, into Euler's formula. F counts every face of the solid, and all 32 panels are faces.
More volume and surface area problems, worked step by step →