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