Polyhedra VEF KEY - Valencia Collegefaculty.valenciacollege.edu/pmurphy2/MGF_1107/Polyhedra VEF KEY.… · considered to be a cube with 6 ... Together with Euler’s famous formula

Polyhedra KEY

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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.

Polyhedra KEY

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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 Square

# of each face 6

# of sides per face 4

total sides of faces 24

TotalSidesofFaces � 2E

24 � 2E

12 � 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 20-

sided 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 Triangle

# of each face 20




60 � 2E

30 � E

V–E�F�2

V–30�20�2

V–10�2

V�12

An icosahedron has 30 edges and 12 vertices.

Polyhedra KEY

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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 Triangle Quadrilateral Hexagon

# of each face 4 5 2

# of sides per face 3 4 6

total sides of faces 12 20 12


12 � 20 � 12 � 2E

44 � 2E

22 � E

V–E�F�2

V–22�11�2

V–11�2

V�13

The convex polyhedron has 22 edges and 13 vertices.

Polyhedra KEY

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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 Triangle Square

# of each face 8 6

# of sides per face 3 4

total sides of faces 24 24


24 � 24 � 2E

48 � 2E

24 � 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.

Polyhedra KEY

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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 Pentagon Triangle Decagon





30 � 30 � 10 � 2E

70 � 2E

35 � E

V–E�F�2

V–35�17�2

V–18�2

V�20

The Pentagonal Rotunda has 35 edges and 20 vertices.

Polyhedra KEY

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6. Below is the net and image for the Augmented Dodecahedron. How many edges and vertices

does it have?

type of face Pentagon Triangle

# of each face 11 5




55 � 15 � 2E

70 � 2E

35 � 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.

Polyhedra KEY

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7. Below is the net for the Pentagonal Orthocupolarontunda. How many edges and vertices does it

have?

type of face Triangle Square Pentagon





45 � 20 � 35 � 2E

100 � 2E

50 � 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.

Polyhedra KEY

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8. Below is the net and image for the Hexagonal Antiprism. How many edges and vertices does it

have?

type of face Triangle Hexagon

# of each face 12 2




36 � 12 � 2E

48 � 2E

24 � 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.

Polyhedra KEY

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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 Triangle

# of each face 8




24 � 2E

12 � E

V–E�F�2

V–12�8�2

V–4�2

V�6

An octahedron has 12 edges and 6 vertices.

Polyhedra KEY

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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 Square Hexagon

# of each face 6 8




24 � 48 � 2E

72 � 2E

36 � 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.

Polyhedra KEY

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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 Pentagon Hexagon

# of each face 12 20




60 � 120 � 2E

180 � 2E

90 � 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.

Polyhedra KEY

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12. The Trapezoidal Hexecontahedron is a polyhedron with 60 quadrilateral faces. How many

vertices and edges does it have?

type of face Quadrilateral

# of each face 60




240 � 2E

120 � 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 Triangle Square Octagon





12 � 52 � 8 � 2E

72 � 2E

36 � 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.

Polyhedra KEY

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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 Decagon Square Triangle Pentagon

# of each face 2 20 10 10

# of sides per face 10 4 3 5

total sides of faces 20 80 30 50


20 � 80 � 30 � 50 � 2E

180 � 2E

90 � 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.

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Polyhedra VEF KEY - Valencia Collegefaculty.valenciacollege.edu/pmurphy2/MGF_1107/Polyhedra VEF KEY.… · considered to be a cube with 6 ... Together with Euler’s famous formula