Polyhedra KEY
Polyhedra (Polyhedron, singular) are 3-D objects with faces that are polygons.
Doing math is just like paradise. ☺
(F) Faces are the flat smooth surfaces. The faces are 2-D polygons.
(E) Edges are where two faces meet.
(V) Vertices are the “points” where edges meet.
NOTE: Remember to keep the vocabulary straight for our formulas! For example, a “six-sided die” is
considered to be a cube with 6 faces, NOT 6 sides!
We will now introduce a new vocabulary phrase “sides of faces”. This is different from a “Face” or from
an “Edge”. When viewing a polygon face, think of it as a 2-D object. Therefore, we can view the sides
of the polygon as “sides of faces”.
“Side of a face” is different than an “Edge” since an “Edge” is a 3-D label. In fact, each “Edge” is
composed of two “sides of faces” because each edge is where two faces meet.
If we go around a polyhedron and count the total number of “sides of faces” from all of its polygon faces,
then we have in fact counted all of the edges of the polyhedron twice! This gives us the following
formula: Total Sides of Faces = 2E. Together with Euler’s famous formula for polyhedra,
V – E + F = 2, we are able to determine the number of edges and vertices of any polyhedron just by
knowing the number and type of polygon faces that it has. NOTE: we consider only convex polyhedra.
1
Polyhedra KEY
Total Sides of Faces = 2E
V–E+F=2
1. A cube has six faces which are all squares. How many edges and vertices does it have?
type of face
# of each face
# of sides per face
total sides of faces
TotalSidesofFaces
24
12
Square
6
4
24
2E
2E
E
V–E F 2
V–12 6 2
V–6 2
V 8
A cube has 12 edges and 8 vertices.
2. Dice are often used in gaming such as the role-playing game Dungeons & Dragons. The 20sided die is a type of polyhedron called an icosahedron. It has 20 faces all of which are
equilateral triangles. How many edges and vertices does an icosahedron have?
type of face
# of each face
# of sides per face
total sides of faces
TotalSidesofFaces
60
30
Triangle
20
3
60
2E
2E
E
V–E F 2
V–30 20 2
V–10 2
V 12
An icosahedron has 30 edges and 12 vertices.
2
Polyhedra KEY
3. Suppose a convex polyhedron has 11 faces, of which 4 are triangles, 5 are quadrilaterals, and 2
are hexagons. How many vertices and edges does it have?
type of face
# of each face
# of sides per face
total sides of faces
TotalSidesofFaces
12 20 12
44
22
Triangle
4
3
12
Quadrilateral
5
4
20
Hexagon
2
6
12
2E
2E
2E
E
V–E F 2
V–22 11 2
V–11 2
V 13
The convex polyhedron has 22 edges and 13 vertices.
3
Polyhedra KEY
4. The cuboctahedron is a polyhedron that can be constructed by slicing off the corners of a cube
through the midpoints of the edges as shown below.
a. How many triangle faces does a cuboctahedron have?
The cuboctahedron has one triangle face for each vertex from the previous cube. Since
the cube has 8 vertices (see question #1) then it follows that the cuboctahedron has
8 triangle faces.
b. How many square faces does a cuboctahedron have?
The cuboctahedron has one square face for each left-over face from the previous cube.
Since the cube has 6 faces (see question #1) then it follows that the cuboctahedron has
6 square faces.
c. How many edges and vertices does a cuboctahedron have?
type of face
# of each face
# of sides per face
total sides of faces
Triangle
8
3
24
TotalSidesofFaces
24 24
48
24
Square
6
4
24
2E
2E
2E
E
NOTE: F 8 6 14
V–E F 2
V–24 14 2
V–10 2
V 12
The cuboctahedron has 24 edges and 12 vertices.
4
Polyhedra KEY
ADDITIONAL EXERCISES
5. A “net” for a polyhedron is a squashed version that shows, in a 2-D
way, how the polygon faces are connected to each other. In fact, if
you were to cut out the net image from this piece of paper, you
could fold it back up into the 3-D polyhedron!
To the right is the net for the Pentagonal Rotunda. You can see
that it has 17 faces, of which 6 are pentagons, 10 are triangles, and
1 is a decagon. How many edges and vertices does it have?
type of face
# of each face
# of sides per face
total sides of faces
TotalSidesofFaces
30 30 10
70
35
Pentagon
6
5
30
Triangle
10
3
30
Decagon
1
10
10
2E
2E
2E
E
V–E F 2
V–35 17 2
V–18 2
V 20
The Pentagonal Rotunda has 35 edges and 20 vertices.
5
Polyhedra KEY
6. Below is the net and image for the Augmented Dodecahedron. How many edges and vertices
does it have?
type of face
# of each face
# of sides per face
total sides of faces
TotalSidesofFaces
55 15
70
35
Pentagon
11
5
55
Triangle
5
3
15
2E
2E
2E
E
NOTE: F 11 5 16
V–E F 2
V–35 16 2
V–19 2
V 21
The Augmented Dodecahedron has 35 edges and 21 vertices.
6
Polyhedra KEY
7. Below is the net for the Pentagonal Orthocupolarontunda. How many edges and vertices does it
have?
type of face
# of each face
# of sides per face
total sides of faces
TotalSidesofFaces
45 20 35
100
50
Triangle
15
3
45
Square
5
4
20
Pentagon
7
5
35
2E
2E
2E
E
NOTE: F 15 5 7 27
V–E F 2
V–50 27 2
V–23 2
V 25
The Pentagonal Orthocupolarontunda has 50 edges and 25 vertices.
7
Polyhedra KEY
8.
Below is the net and image for the Hexagonal Antiprism. How many edges and vertices does it
have?
type of face
# of each face
# of sides per face
total sides of faces
TotalSidesofFaces
36 12
48
24
Triangle
12
3
36
Hexagon
2
6
12
2E
2E
2E
E
NOTE: F 12 2 14
V–E F 2
V–24 14 2
V–10 2
V 12
The Hexagonal Antiprism has 24 edges and 12 vertices.
8
Polyhedra KEY
9. The 8-sided die is a type of polyhedron called an octahedron. It is a polyhedron with eight
triangle faces. How many vertices and edges does it have?
type of face
# of each face
# of sides per face
total sides of faces
TotalSidesofFaces
24
12
Triangle
8
3
24
2E
2E
E
V–E F 2
V–12 8 2
V–4 2
V 6
An octahedron has 12 edges and 6 vertices.
9
Polyhedra KEY
10. The truncated octahedron can be constructed by slicing off the corners of an octahedron so that a
square is formed.
a. How many square faces does a truncated octahedron have?
The truncated octahedron has one square face for each vertex from the original
octahedron. Since an octahedron has 6 vertices (see question #9) then it follows that the
truncated octahedron has 6 square faces.
b. How many hexagon faces does a truncated octahedron have?
The truncated octahedron has one hexagon face for each left-over face from the original
octahedron. Since an octahedron has 8 faces (see question #9) then it follows that the
truncated octahedron has 8 hexagon faces.
c. How many edges and vertices does a truncated octahedron have?
type of face
# of each face
# of sides per face
total sides of faces
TotalSidesofFaces
24 48
72
36
Square
6
4
24
Hexagon
8
6
48
2E
2E
2E
E
NOTE: F 6 8 14
V–E F 2
V–36 14 2
V–22 2
V 24
The truncated octahedron has 36 edges and 24 vertices.
10
Polyhedra KEY
11. The truncated icosahedron can be constructed by slicing off the corners of an icosahedron so that
a pentagon is formed.
a. How many pentagon faces does the truncated icosahedron have?
The truncated icosahedron has one pentagon face for each vertex from the original
icosahedron. Since an icosahedron has 12 vertices (see question #2) then it follows that
the truncated icosahedron has 12 pentagon faces.
b. How many hexagon faces does the truncated icosahedron have?
The truncated icosahedron has one hexagon face for each left-over face from the original
icosahedron. Since an icosahedron has 20 faces (see question #2) then it follows that the
truncated icosahedron has 20 hexagon faces.
c. How many edges and vertices does the truncated icosahedron have?
type of face
# of each face
# of sides per face
total sides of faces
TotalSidesofFaces
60 120
180
90
Pentagon
12
5
60
Hexagon
20
6
120
2E
2E
2E
E
NOTE: F 12 20 32
V–E F 2
V–90 32 2
V–58 2
V 60
The truncated icosahedron has 90 edges and 60 vertices.
11
Polyhedra KEY
12. The Trapezoidal Hexecontahedron is a polyhedron with 60 quadrilateral faces. How many
vertices and edges does it have?
type of face
# of each face
# of sides per face
total sides of faces
Quadrilateral
60
4
240
TotalSidesofFaces
240
120
2E
2E
E
V–E F 2
V–120 60 2
V–60 2
V 62
The Trapezoidal Hexecontahedron has 120 edges and 62 vertices.
13. The Elongated Square Cupola is a polyhedron with the following faces: 4 triangles, 13 squares,
and 1 octagon. How many edges and vertices does it have?
type of face
# of each face
# of sides per face
total sides of faces
Triangle
4
3
12
TotalSidesofFaces
12 52 8
72
36
Square
13
4
52
Octagon
1
8
8
2E
2E
2E
E
NOTE: F 4 13 1 18
V–E F 2
V–36 18 2
V–18 2
V 20
The Elongated Square Cupola has 36 edges and 20 vertices.
12
Polyhedra KEY
14. The Gyrate Bidiminished Rhombicosidodecahedron is a polyhedron with the following faces:
2 decagons, 20 squares, 10 triangles, and 10 pentagons. How many vertices and edges does it
have?
type of face
# of each face
# of sides per face
total sides of faces
Decagon
2
10
20
TotalSidesofFaces
20 80 30 50
180
90
Square
20
4
80
Triangle
10
3
30
Pentagon
10
5
50
2E
2E
2E
E
NOTE: F 2 20 10 10 42
V–E F 2
V–90 42 2
V–48 2
V 50
The Gyrate Bidiminished Rhombicosidodecahedron has 90 edges and 50 vertices.
13