Euler’s Formula
A Naturally Occurring Function
Leonhard Euler was a brilliant
Swiss mathematician. He is
often referred to as the
“Beethoven of Mathematics.”
Euler discovered an interesting
relationship between the number
of faces, vertices, and edges for
any polyhedron.
Poly-what?
A polyhedron is a 3 dimensional
shape with flat sides.
All prisms and pyramids are
examples of polyhedra (plural for
polyhedron).
POLYHEDRA
PRISMS
PYRAMIDS
Any polyhedron has faces,
vertices, and edges.
EDGE
FACE
VERTEX
A face is a flat side.
This rectangular prism has 6
faces.
This rectangular prism has 6
faces.
FRONT
This rectangular prism has 6
faces.
BACK
FRONT
This rectangular prism has 6
faces.
TOP
BACK
FRONT
This rectangular prism has 6
faces.
TOP
BACK
FRONT
BOTTOM
This rectangular prism has 6
faces.
TOP
BACK
FRONT
L
E
F
T
BOTTOM
This rectangular prism has 6
faces.
TOP
BACK
FRONT
R
I
G
H
T
L
E
F
T
BOTTOM
This rectangular prism has 6
faces.
TOP
BACK
FRONT
R
I
G
H
T
L
E
F
T
BOTTOM
This rectangular prism has 6
faces.
TOP
BACK
FRONT
R
I
G
H
T
L
E
F
T
BOTTOM
This rectangular prism has 6
faces.
L
E
F
T
TOP
BACK
FRONT
R
I
G
H
T
BOTTOM
This rectangular prism has 6
faces.
L
E
F
T
TOP
BACK
FRONT
BOTTOM
R
I
G
H
T
This rectangular prism has 6
faces.
L
E
F
T
TOP
BACK
FRONT
BOTTOM
R
I
G
H
T
This square pyramid has 5 faces.
The faces consist of 4 triangles and a square.
The faces consist of 4 triangles and a square.
A triangular pyramid has 4 faces.
A triangular pyramid has 4 faces.
A triangular pyramid has 4 faces.
A triangular pyramid has 4 faces.
A triangular pyramid has 4 faces.
A triangular pyramid has 4 faces.
An edge is a line segment where
two faces meet.
A rectangular prism has 12
edges.
A triangular pyramid has 6 edges.
A vertex is a corner.
It is a point that connects 2 or
more edges.
A vertex is a fancy word for
“corner.”
C
A
B
Every triangle has 3 vertices (corners).
Points A, B, and C are vertices.
A rectangular prism has 8
vertices.
A rectangular prism has 8
vertices.
A triangular pyramid has 4
vertices.
A triangular pyramid has 4
vertices.
Euler studied the faces, vertices,
and edges of different polyhedra.
Like most great mathematicians
and scientists, he organized his
data in a chart.
Polyhedron
# of Faces
# of Vertices
# of Edges
Cube
6
8
12
Sq. Pyramid
5
5
8
Tri. Prism
5
6
9
Euler looked for a relationship
between these numbers.
Polyhedron
# of Faces
# of Vertices
# of Edges
Cube
6
8
12
Sq. Pyramid
5
5
8
Tri. Prism
5
6
9
Can you determine Euler’s formula that
relates the # of Faces and # of Vertices to
the # of Edges?
Polyhedron
# of Faces
# of Vertices
# of Edges
Cube
6
8
12
Sq. Pyramid
5
5
8
Tri. Prism
5
6
9
Faces + Vertices –2 = Edges
Polyhedron
# of Faces
Cube
6
Sq. Pyramid
5
Tri. Prism
5
# of Vertices
+
+
+
8
5
6
# of Edges
-2=
-2=
-2=
12
8
9
Use Euler’s Formula to
determine the number of edges in
a pentagonal prism.
Polyhedron
# of Faces
Cube
6
Sq. Pyramid
5
Tri. Prism
5
Pent. Prism
7
# of Vertices
+
+
+
8
5
6
10
# of Edges
-2=
-2=
-2=
12
8
9
Use Euler’s Formula to
determine the number of edges in
a pentagonal prism.
Polyhedron
# of Faces
Cube
6
Sq. Pyramid
5
Tri. Prism
5
Pent. Prism
7
# of Vertices
+
+
+
+
8
5
6
10
# of Edges
-2=
-2=
-2=
-2=
12
8
9
Use Euler’s Formula to
determine the number of edges in
a pentagonal prism.
Polyhedron
# of Faces
Cube
6
Sq. Pyramid
5
Tri. Prism
5
Pent. Prism
7
# of Vertices
+
+
+
+
8
5
6
10
# of Edges
-2=
-2=
-2=
-2=
12
8
9
15
SUMMARY:
Euler’s Formula says that if you
add the number of faces and
vertices, then subtract by 2, the
result is the number of edges.
Euler’s Formula works for any
polyhedron.
THE END!