1. Explore Solids
Warm Up
Lesson Presentation
Lesson Quiz

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

3. Warm-Up
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

4. Example 1
Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. a.
SOLUTION
a. 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.

5. Example 1
Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. b.
SOLUTION
b. 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.

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

7. Guided Practice
Describe the solid. If it is a polyhedron, 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, consisting of 1 base and 4 sides. It has 5 vertices and 8 edges.
ANSWER

8. Guided Practice
Describe the solid. If it is a polyhedron, find the number of faces, vertices, and edges. 2.
It has a curved surface, so it is not a polyhedron.
ANSWER

9. Guided Practice
Describe the solid. If it is a polyhedron, find the number of faces, vertices, and edges. 3.
The solid is formed by polygons, so it is a polyhedron. The two bases are triangles, so it is a triangular prism. It has 5 faces, consisting of 2 triangular bases and 3 rectangular faces. It has 6 vertices and 9 edges.
ANSWER

10. Example 2
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.
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.

11. Use Euler’s Theorem to find the number of edges.
Example 2
Use Euler’s Theorem to find the number of edges.
F + V = E + 2
Euler’s Theorem
= E + 2
Substitute known values.
15 = E
Solve for E.
The frame of the house has 15 edges.

12. Example 3
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
Euler’s Theorem =
Substitute.
14 = 14
This is a true statement. So, the solution checks.

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

14. Example 4
Describe the shape formed by the intersection of the
plane and the cube. b.
SOLUTION
b. The cross section is a rectangle.

15. Example 4
Describe the shape formed by the intersection of the
plane and the cube. c.
SOLUTION
c. The cross section is a trapezoid.

16. Guided Practice
4. Find the number of faces, vertices, and edges of the regular dodecahedron on page 828. Check your answer using Euler’s Theorem.
ANSWER
edges: 30; F + V = E + 2, = , 32 = 32

17. Guided Practice
5. Identify and sketch the cross section of the prism. Then find its perimeter and area. The base of the prism is a regular polygon with sides 6 meters long.
ANSWER
Regular hexagon; check sketches: P = 36 m; A =  93.5 m2

18. Lesson Quiz
Determine whether the solid is a polyhedron. If it is, name it. 1.
ANSWER no

19. Lesson Quiz
Determine whether the solid is a polyhedron. If it is, name it. 2.
ANSWER
yes; pyramid

20. Lesson Quiz
Determine whether the solid is a polyhedron. If it is, name it. 3.
ANSWER
yes; pentagonal prism

21. Lesson 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