1. Name the polygon by the number of sides.
3. 5
1. 6
2. 10
ANSWER
hexagon
ANSWER
decagon
ANSWER
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

2. 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.

3. 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.

4. 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

5. EXAMPLE 1
Identify and name polyhedra
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.

6. EXAMPLE 1
Identify and name polyhedra
Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges.
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.
SOLUTION

7. EXAMPLE 1
Identify and name polyhedra
Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges.
The cone has a curved surface, so it is not a
polyhedron.
SOLUTION

8. 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.
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.
ANSWER

9. 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.
It has a curved side, so it is not a polyhedron.
ANSWER

10. 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.
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.
ANSWER

11. EXAMPLE 2
Use Euler’s Theorem in a real-world situation
Find the number of edges on the frame of the house.
House Construction
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.

12. Use Euler’s Theorem in a real-world situation
EXAMPLE 2
Use Euler’s Theorem in a real-world situation
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
= E + 2
15 = E
Euler’s Theorem
Substitute known values.
Solve for E.
The frame of the house has 15 edges.
ANSWER

13. Use Euler’s Theorem with Platonic solids
EXAMPLE 3
Use Euler’s Theorem with Platonic solids
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.
F + V = E + 2 =
14 = 14
Euler’s Theorem
Substitute.
This is a true statement. So, the solution checks.

14. EXAMPLE 4
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.

15. EXAMPLE 4
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.

16. EXAMPLE 4
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.

17. 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

18. 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

19. 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

20. Daily Homework Quiz
Determine whether the solid is a polyhedron. If it is, name it. 1. 2. 3.
ANSWER
yes; pentagonal prism
ANSWER no
ANSWER
yes; pyramid

21. Daily Homework Quiz
4. Find the number of faces, vertices, and edges
of each polyhedron in Exercises 1–3. 1. 2. 3.
ANSWER
pyramid: 5 faces, 5 vertices, 8 edges;
prism: 7 faces, 10 vertices, 15 edges

22. 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