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

2. Name the polygon by the number of sides.
3. 5
ANSWER
pentagon
4. What is a regular polygon?
ANSWER
a polygon with all congruent sides and all congruent angles

3. Name the polygon by the number of sides.
5. What is the length of the diagonal of a square with side length 6?
ANSWER
6 2

4. 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. a. b.

5. EXAMPLE 1
Identify and name polyhedron c.
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. a.

6. EXAMPLE 1
Identify and name polyhedron
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. b.
The cone has a curved surface, so it is not a
polyhedron. c.

7. 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 it has 5 faces, 5 vertices and 8 edges.
ANSWER

8. GUIDED PRACTICE
for Example 1 2.
It has a curved square, so it is not a polyhedron.
ANSWER

9. GUIDED PRACTICE
for Example 1 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 face, 6 vertices, and 9 edges.
ANSWER

10. EXAMPLE 2
Use Euler’s Theorem in a real-world situation
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.

11. 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

12. 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.

13. EXAMPLE 4
Describe cross sections
Describe the shape formed by the intersection of the
plane and the cube. a. b.
SOLUTION
a. The cross section is a square.
b. The cross section is a rectangle.

14. EXAMPLE 4
Describe cross sections c.
SOLUTION
c. The cross section is a trapezoid.

15. GUIDED PRACTICE for Examples 2, 3, and 4
4. Find the number of faces, vertices, and edges of the regular dodecahedron on page 796. Check your answer using Euler’s Theorem.
SOLUTION
Counting on the diagram, the dodecahedron has 12 faces, 20 vertices, and 30 edges. Use Euler’s theorem to
F + V = E + 2 =
32 = 32
Check
Euler’s theorem
Substitute
This is a true statement so, the solution check

16. 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

17. GUIDED PRACTICE
for Examples 2, 3, and 4 6.
ANSWER
The cross section is a circle

18. GUIDED PRACTICE
for Examples 2, 3, and 4 7.
ANSWER
The cross section is a hexagon

19. Daily Homework Quiz
Determine whether the solid is a polyhedron. If it is, name it. 1.
ANSWER no

20. Daily Homework Quiz 2.
ANSWER
yes; pyramid

21. Daily Homework Quiz 3.
ANSWER
yes; pentagonal prism

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