warm-up exercises 1.6 2.10 answer hexagon answer decagon name the polygon by the number of sides

Warm-Up Exercises

1. 6

2. 10

ANSWER hexagon

ANSWER decagon

Name the polygon by the number of sides.

Warm-Up Exercises

3. 5

4. What is a regular polygon?

ANSWER a polygon with all congruent sides and allcongruent angles

ANSWER pentagon

Name the polygon by the number of sides.

Warm-Up Exercises

5. What is the length of the diagonal of a square withside length 6?

Name the polygon by the number of sides.

ANSWER 6 2

Warm-Up ExercisesEXAMPLE 1

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

a. b.

Identify and name polyhedra

Warm-Up ExercisesEXAMPLE 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.

Warm-Up ExercisesEXAMPLE 1

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.

Identify and name polyhedron

Warm-Up ExercisesGUIDED 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

Warm-Up ExercisesGUIDED PRACTICE for Example 1

2.

It has a curved square, so it is not a polyhedron.

ANSWER

Warm-Up ExercisesGUIDED 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

Warm-Up ExercisesEXAMPLE 2 Use Euler’s Theorem in a real-world situation

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.

Find the number of edges on the frame of the house.

House Construction

Warm-Up ExercisesEXAMPLE 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

7 + 10 = E + 2

15 = E

Euler’s Theorem

Substitute known values.

Solve for E.

The frame of the house has 15 edges.ANSWER

Warm-Up ExercisesEXAMPLE 3 Use Euler’s Theorem with Platonic solids

SOLUTION

By counting on the diagram, the octahedron has 8 faces, 6 vertices, and 12 edges. Use Euler’s Theorem to check.

Find the number of faces, vertices, and edges of the regular octahedron. Check your answer using Euler’s Theorem.

F + V = E + 2

8 + 6 = 12 + 2

14 = 14

Euler’s Theorem

Substitute.

This is a true statement. So, the solution checks.

Warm-Up ExercisesEXAMPLE 4

Describe the shape formed by the intersection of theplane and the cube.

a. b.

Describe cross sections

b. The cross section is a rectangle.

a. The cross section is a square.

SOLUTION

Warm-Up ExercisesEXAMPLE 4 Describe cross sections

SOLUTION

c. The cross section is a trapezoid.

c.

Warm-Up ExercisesGUIDED 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.

Counting on the diagram, the dodecahedron has 12 faces, 20 vertices, and 30 edges. Use Euler’s theorem to

F + V = E + 2

12 + 20 = 30 + 2

32 = 32

Check Euler’s theorem

Substitute

This is a true statement so, the solution check

SOLUTION

Warm-Up ExercisesGUIDED PRACTICE for Examples 2, 3, and 4

Describe the shape formed by the intersection of theplane and the solid.

5.

ANSWER

The cross section is a triangle

Warm-Up ExercisesGUIDED PRACTICE for Examples 2, 3, and 4

6.

ANSWER

The cross section is a circle

Warm-Up ExercisesGUIDED PRACTICE for Examples 2, 3, and 4

7.

ANSWER

The cross section is a hexagon

Warm-Up ExercisesDaily Homework Quiz

Determine whether the solid is a polyhedron. If it is, name it.

1.

ANSWER

no

Warm-Up ExercisesDaily Homework Quiz

2.

ANSWER

yes; pyramid

Warm-Up ExercisesDaily Homework Quiz

3.

ANSWER

yes; pentagonal prism

Warm-Up ExercisesDaily Homework Quiz

4. Find the number of faces, vertices, and edges

of each polyhedron in Exercises 1–3.

5. A plane intersects a cone, but does not intersect the base of the cone. Describe the possible cross sections.

ANSWER

pyramid: 5 faces, 5 vertices, 8 edges;prism: 7 faces, 10 vertices, 15 edges

ANSWER

a point, a circle, an ellipse