You will need
11.6 Polyhedron Faces, Edges,
• pipe cleaners or
straws
• modelling clay
and Vertices
GOAL
Determine how the number of faces, edges, and vertices of a polyhedron are related.
Learn about the Math
Toma and Benjamin noticed that whenever you make a 2-D polygon, the
number of vertices is the same as the number of edges. They wondered
whether the number of faces, vertices, and edges of 3-D polyhedrons are
related.
pattern links the number of faces, edges, and
? What
vertices of a polyhedron?
Benjamin tried building an unusual polyhedron first. “I’ll start building it
from the top. The number of vertices and number of edges are the same,
and there is one face.”
Part built
Number of faces
Number of vertices
top
1
4
390 Chapter 11
Number of edges
4
NEL
“Next, I’ll add squares. For each square, I add one new face, two new
vertices, and three new edges. The total number of new faces and new
vertices is equal to the number of new edges.
“Now I’ll add triangles. For each triangle, I add one new face and one
new edge, but no new vertices. Again, the total number of new faces and
vertices is the same as the number of new edges.”
Part built
top
4 new squares
4 new triangles
Number of
faces
Number of
vertices
Number of
edges
1
4 more
4 more
4
8 more
0 more
4
12 more
4 more
A. Construct the parts that Benjamin constructed. Add the next set of
squares. Explain why the number of edges is 1 less than the total
number of faces and vertices.
B. Add the bottom of the polyhedron. Explain why you have added no
new edges or vertices, but one new face.
C. Why is the number of edges 2 less than the total number of faces and
vertices?
D. Choose one of the following shapes. Compare the number of edges
with the total number of faces and vertices. What do you observe?
E. Compare your results with the results of students who chose different
shapes. What do you notice?
Reflecting
1. The relationship you described in step C is called Euler’s formula
(pronouced “oiler”). Explain why it can be written as F V E 2,
where F is the number of faces, V is the number of vertices, and E is
the number of edges of the shape.
2. How does Euler’s formula allow you to predict the number of edges,
faces, or vertices of a shape if you know two of these values?
NEL
Geometry and Measurement Relationships
391
Work with the Math
Example 1: Checking whether a polyhedron is possible
Is it possible to make a polyhedron with 6 faces, 7 vertices, and 10 edges?
Tran’s Solution
FVE2
F V E 6 7 10
3
I used Euler’s formula. If it is possible to make a polyhedron
like this, the result should be 2 when I substitute the values into
Euler’s formula.
I substituted the values into the formula. The result is 3, not 2,
so it is not possible to make such a polyhedron.
Example 2: Using Euler’s formula to determine a missing value
If a polyhedron has 10 faces and 18 edges, how many vertices should it have?
Benjamin’s Solution
FVE2
10 V 18 2
V82
V8828
V 10
A
I used Euler’s formula.
I substituted 10 for the number of faces and 18 for the number
of edges.
I used balancing to solve the equation.
The polyhedron should have 10 vertices.
Checking
3. A student used 10 pipe cleaners to make the
edges of a polyhedron. If the polyhedron has
6 vertices, how many faces must it have?
4. Show that Euler’s formula works for a
tetrahedron.
392 Chapter 11
B
Practising
5. Show that Euler’s formula works for the
other four Platonic solids: a cube, an
octahedron, a dodecahedron, and an
icosahedron.
NEL
6. Copy and complete the chart for some
polyhedrons.
Number of
faces
Number of
edges
Number of
vertices
9
12
5
6
6
20
16
7
10. Make another cube using modelling clay.
Then make a pyramid on each face of the
cube. Show that Euler’s formula works for
this polyhedron.
11. Imagine that you drilled a rectangular hole
through a cube. Does Euler’s formula work
for the new shape?
30
12
10
6
12. a) Construct a triangular prism.
b)
c)
d)
e)
7. The following crystals and gemstones have
been cut to form polyhedrons. Show that
Euler’s formula works for each polyhedron.
a)
How many faces does the prism have?
How many edges does the prism have?
How many vertices does the prism have?
Show that Euler’s formula works for
the prism.
13. Repeat question 12 using a pentagonal
pyramid.
C
b)
Extending
14. Make a cube using modelling clay. Mark a
point in the centre of each face. Imagine that
you joined these points with string inside the
cube to form a polyhedron. Show that
Euler’s formula works for this polyhedron.
15. A prism has a base with n sides.
8. Show that Euler’s
formula works for
this cuboctahedron.
a)
b)
c)
d)
How many faces does the prism have?
How many edges does the prism have?
How many vertices does the prism have?
Show that Euler’s formula works for
the prism.
16. A pyramid has a base with n sides.
9. Make a cube using
modelling clay. Cut
the corners off the
cube. Show that
Euler’s formula
works for the new
shape.
NEL
a) How many faces does the pyramid
have?
b) How many edges does the pyramid
have?
c) How many vertices does the pyramid
have?
d) Show that Euler’s formula works for
the pyramid.
Geometry and Measurement Relationships
393