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Warm-Up Exercises
Name the polygon by the number of sides.
1. 6
2. 10
3. 5
ANSWER
ANSWER
ANSWER
hexagon
decagon
pentagon
4. What is a regular polygon?
ANSWER
a polygon with all congruent sides and all
congruent angles
5. What is the length of the diagonal of a square with
side length 6?
ANSWER 6 2
Warm-Up Exercises
Vocabulary
• Polyhedron – is a solid bounded by polygons (called
faces) that enclose a single region of space.
The line segment formed by
the intersection of two faces
is called an edge.
Three or more faces intersect
in a single point called a
vertex.
Bases are two parallel,
congruent faces. The solids
are named by the shape of
their base.
Warm-Up Exercises
Vocabulary
A polyhedron is regular if all
of its faces are congruent
regular polygons.
A polyhedron is convex if the
segment joining any two
points on the surface lies
entirely on the surface and in
the interior region. If the
polygon is not convex, then it
is concave.
Warm-Up Exercises
Vocabulary
• Platonic Solids – The five regular polyhedron.
• Euler’s Theorem – The number of faces (F), vertices
(V), and edges (E) of a polyhedron are related by the
formula
F+V=E+2
http://www.learner.org/interactives/geometry/platonic.html
Identify and name polyhedra
Warm-Up1Exercises
EXAMPLE
Tell whether the solid is a polyhedron. If it is, name the
polyhedron and find the number of faces, vertices, and
edges.
SOLUTION
The solid is formed by
polygons, so it is a
polyhedron. The two bases are
congruent rectangles, so it is a
rectangular prism. It has 6
faces, 8 vertices, and 12 edges.
Identify and name polyhedra
Warm-Up1Exercises
EXAMPLE
Tell whether the solid is a polyhedron. If it is, name the
polyhedron and find the number of faces, vertices, and
edges.
SOLUTION
The solid is formed by polygons,
so it is a polyhedron. The base is a
hexagon, so it is a hexagonal
pyramid. It has 7 faces, consisting
of 1 base, 3 visible triangular faces,
and 3 non-visible triangular faces.
The polyhedron has 7 faces, 7
vertices, and 12 edges.
Identify and name polyhedra
Warm-Up1Exercises
EXAMPLE
Tell whether the solid is a polyhedron. If it is, name the
polyhedron and find the number of faces, vertices, and
edges.
SOLUTION
The cone has a curved
surface, so it is not a
polyhedron.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1
Tell whether the solid is a polyhedron. If it is, name
the polyhedron and find the number of faces,
vertices, and edges.
1.
ANSWER
The solid is formed by
polygons so it is a
polyhedron. The base is
a square, so it is a
square pyramid. It has 5
faces, 5 vertices and 8
edges.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1
Tell whether the solid is a polyhedron. If it is, name
the polyhedron and find the number of faces,
vertices, and edges.
2.
ANSWER
It has a curved side, so it is
not a polyhedron.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1
Tell whether the solid is a polyhedron. If it is, name
the polyhedron and find the number of faces,
vertices, and edges.
3.
ANSWER
The solid is formed by
polygons, so it is a
polyhedron. The base is a
triangle, so it is a
triangular prism. It has 5
faces, 6 vertices, and 9
edges.
Use Euler’s Theorem in a real-world situation
Warm-Up2Exercises
EXAMPLE
House Construction
Find the number of
edges on the frame of
the house.
SOLUTION
The frame has one face as its
foundation, four that make up its walls, and two that
make up its roof, for a total of 7 faces.
Use Euler’s Theorem in a real-world situation
Warm-Up2Exercises
EXAMPLE
To find the number of vertices,
notice that there are 5 vertices
around each pentagonal wall, and
there are no other vertices. So, the
frame of the house has 10 vertices.
Use Euler’s Theorem to find the number of edges.
F +V = E+2
Euler’s Theorem
7 + 10 = E + 2
Substitute known values.
15 = E
Solve for E.
ANSWER The frame of the house has 15 edges.
Use Euler’s Theorem with Platonic solids
Warm-Up3Exercises
EXAMPLE
Find the number of faces, vertices, and
edges of the regular octahedron.
Check your answer using Euler’s
Theorem.
SOLUTION
By counting on the diagram, the octahedron has 8
faces, 6 vertices, and 12 edges. Use Euler’s Theorem
to check.
Euler’s Theorem
F+V = E+2
8 + 6 = 12 + 2
14 = 14
Substitute.
This is a true statement. So, the solution
checks.
Warm-Up4Exercises
EXAMPLE
Describe cross sections
Describe the shape formed by the intersection of the
plane and the cube.
a.
SOLUTION
a.
The cross section is a square.
Warm-Up4Exercises
EXAMPLE
Describe cross sections
Describe the shape formed by the intersection of the
plane and the cube.
b.
SOLUTION
b.
The cross section is a rectangle.
Warm-Up4Exercises
EXAMPLE
Describe cross sections
Describe the shape formed by the intersection of the
plane and the cube.
c.
SOLUTION
c.
The cross section is a trapezoid.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2, 3, and 4
Describe the shape formed by the intersection of the
plane and the solid.
5.
ANSWER
The cross section is a triangle
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2, 3, and 4
Describe the shape formed by the intersection of the
plane and the solid.
6.
ANSWER
The cross section is a circle
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2, 3, and 4
Describe the shape formed by the intersection of the
plane and the solid.
7.
ANSWER
The cross section is a hexagon
Warm-Up
Exercises
Daily
Homework
Quiz
Determine whether the solid is a polyhedron. If it is, name it.
1.
2.
3.
ANSWER
ANSWER
ANSWER
no
yes; pyramid
yes;
pentagonal
prism
Warm-Up
Exercises
Daily
Homework
Quiz
4.
1.
Find the number of faces, vertices, and edges
of each polyhedron in Exercises 1–3.
2.
3.
ANSWER
pyramid: 5 faces, 5 vertices, 8 edges;
prism: 7 faces, 10 vertices, 15 edges
Warm-Up
Exercises
Daily
Homework
Quiz
5.
A plane intersects a cone, but does not
intersect the base of the cone. Describe the
possible cross sections.
ANSWER
a point, a circle, an ellipse