You will need 11.6 Polyhedron Faces, Edges, • modelling...
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What pattern links the number of faces, edges, and 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.” 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. 390 Chapter 11 NEL GOAL Determine how the number of faces, edges, and vertices of a polyhedron are related. Learn about the Math You will need • pipe cleaners or straws • modelling clay Polyhedron Faces, Edges, and Vertices 11.6 ? Part built Number of faces Number of vertices Number of edges top 1 4 4
Transcript of You will need 11.6 Polyhedron Faces, Edges, • modelling...
What pattern links the number of faces, edges, andvertices of a polyhedron?
Benjamin tried building an unusual polyhedron first. “I’ll start building itfrom the top. The number of vertices and number of edges are the same,and there is one face.”
Toma and Benjamin noticed that whenever you make a 2-D polygon, thenumber of vertices is the same as the number of edges. They wonderedwhether the number of faces, vertices, and edges of 3-D polyhedrons arerelated.
390 Chapter 11 NEL
GOALDetermine how the number of faces, edges, and vertices of a polyhedron are related.
Learn about the Math
You will need• pipe cleaners or
straws• modelling clayPolyhedron Faces, Edges,
and Vertices11.6
?
Part built Number of faces Number of vertices Number of edges
top 1 4 4
E. Compare your results with the results of students who chose differentshapes. What do you notice?
Reflecting1. 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 isthe 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?
A. Construct the parts that Benjamin constructed. Add the next set ofsquares. Explain why the number of edges is 1 less than the totalnumber of faces and vertices.
B. Add the bottom of the polyhedron. Explain why you have added nonew edges or vertices, but one new face.
C. Why is the number of edges 2 less than the total number of faces andvertices?
D. Choose one of the following shapes. Compare the number of edgeswith the total number of faces and vertices. What do you observe?
NEL Geometry and Measurement Relationships 391
“Next, I’ll add squares. For each square, I add one new face, two newvertices, and three new edges. The total number of new faces and newvertices is equal to the number of new edges.
“Now I’ll add triangles. For each triangle, I add one new face and onenew edge, but no new vertices. Again, the total number of new faces andvertices is the same as the number of new edges.”
Part built
Number of
faces
Number of
vertices
Number of
edges
top 1 4 44 new squares 4 more 8 more 12 more4 new triangles 4 more 0 more 4 more
NEL392 Chapter 11
Checking3. A student used 10 pipe cleaners to make the
edges of a polyhedron. If the polyhedron has6 vertices, how many faces must it have?
4. Show that Euler’s formula works for atetrahedron.
A Practising5. Show that Euler’s formula works for the
other four Platonic solids: a cube, anoctahedron, a dodecahedron, and anicosahedron.
B
Example 2: Using Euler’s formula to determine a missing value
I used Euler’s formula.
I substituted 10 for the number of faces and 18 for the number of edges.
If a polyhedron has 10 faces and 18 edges, how many vertices should it have?
Benjamin’s Solution
F � V � E � 2
10 � V � 18 � 2
I used balancing to solve the equation.
The polyhedron should have 10 vertices.
V � 8 � 2
V � 8 � 8 � 2 � 8
V � 10
Work with the Math
Example 1: Checking whether a polyhedron is possible
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 intoEuler’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.
Is it possible to make a polyhedron with 6 faces, 7 vertices, and 10 edges?
Tran’s Solution
F � V � E � 2
F � V � E � 6 � 7 � 10� 3
NEL Geometry and Measurement Relationships 393
9. Make a cube usingmodelling clay. Cutthe corners off thecube. Show thatEuler’s formulaworks for the newshape.
10. Make another cube using modelling clay.Then make a pyramid on each face of thecube. Show that Euler’s formula works forthis polyhedron.
11. Imagine that you drilled a rectangular holethrough a cube. Does Euler’s formula workfor the new shape?
12. a) Construct a triangular prism.
b) How many faces does the prism have?
c) How many edges does the prism have?
d) How many vertices does the prism have?
e) Show that Euler’s formula works forthe prism.
13. Repeat question 12 using a pentagonalpyramid.
Extending14. Make a cube using modelling clay. Mark a
point in the centre of each face. Imagine thatyou joined these points with string inside thecube to form a polyhedron. Show thatEuler’s formula works for this polyhedron.
15. A prism has a base with n sides.
a) How many faces does the prism have?
b) How many edges does the prism have?
c) How many vertices does the prism have?
d) Show that Euler’s formula works forthe prism.
16. A pyramid has a base with n sides.
a) How many faces does the pyramidhave?
b) How many edges does the pyramidhave?
c) How many vertices does the pyramidhave?
d) Show that Euler’s formula works forthe pyramid.
C
6. Copy and complete the chart for somepolyhedrons.
7. The following crystals and gemstones havebeen cut to form polyhedrons. Show thatEuler’s formula works for each polyhedron.
a)
b)
8. Show that Euler’sformula works forthis cuboctahedron.
Number of
faces
Number of
edges
Number of
vertices
9 56 126 7
20 3016 10
12 6
