Download 11.4 Three-Dimensional Figures and more Study notes Reasoning in PDF only on Docsity!
Section 11.4 Three-Dimensional Figures 617
11.4 Three-Dimensional Figures
Essential QuestionEssential Question What is the relationship between the numbers
of vertices V , edges E , and faces F of a polyhedron?
A polyhedron is a solid that is bounded by polygons, called faces.
- Each vertex is a point.
- Each edge is a segment of a line.
- Each face is a portion of a plane.
Analyzing a Property of Polyhedra
Work with a partner. The five Platonic solids are shown below. Each of these solids has congruent regular polygons as faces. Complete the table by listing the numbers of vertices, edges, and faces of each Platonic solid.
tetrahedron cube octahedron
dodecahedron icosahedron
Solid Vertices, V Edges, E Faces, F
tetrahedron
cube
octahedron
dodecahedron
icosahedron
Communicate Your AnswerCommunicate Your Answer
2. What is the relationship between the numbers of vertices V , edges E , and faces F of a polyhedron? ( Note : Swiss mathematician Leonhard Euler (1707–1783) discovered a formula that relates these quantities.) 3. Draw three polyhedra that are different from the Platonic solids given in Exploration 1. Count the numbers of vertices, edges, and faces of each polyhedron. Then verify that the relationship you found in Question 2 is valid for each polyhedron.
CONSTRUCTING
VIABLE ARGUMENTS
To be proficient in math, you need to reason inductively about data.
edge
face
vertex
618 Chapter 11 Circumference, Area, and Volume
11.4 Lesson^ What You Will LearnWhat You Will Learn
Classify solids. Describe cross sections. Sketch and describe solids of revolution.
Classifying Solids A three-dimensional figure, or solid, is bounded by fl at or curved surfaces that enclose a single region of space. A polyhedron is a solid that is bounded by polygons, called faces. An edge of a polyhedron is a line segment formed by the intersection of two faces. A vertex of a polyhedron is a point where three or more edges meet. The plural of polyhedron is polyhedra or polyhedrons.
To name a prism or a pyramid, use the shape of the base. The two bases of a prism are congruent polygons in parallel planes. For example, the bases of a pentagonal prism are pentagons. The base of a pyramid is a polygon. For example, the base of a triangular pyramid is a triangle.
Classifying Solids
Tell whether each solid is a polyhedron. If it is, name the polyhedron. a. b. c.
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. b. The solid is formed by polygons, so it is a polyhedron. The base is a hexagon, so it is a hexagonal pyramid. c. The cone has a curved surface, so it is not a polyhedron.
polyhedron, p. 618 face, p. 618 edge, p. 618 vertex, p. 618 cross section, p. 619 solid of revolution, p. 620 axis of revolution, p. 620 Previous solid prism pyramid cylinder cone sphere base
Core VocabularyCore Vocabullarry
CoreCore ConceptConcept
Types of Solids
prism
pyramid
Polyhedra
cylinder cone
sphere
Not Polyhedra
face
edge
vertex
Pentagonal prism
Bases are pentagons.
Triangular pyramid
Base is a triangle.
620 Chapter 11 Circumference, Area, and Volume
Sketching and Describing Solids of Revolution A solid of revolution is a three-dimensional figure that is formed by rotating a two-dimensional shape around an axis. The line around which the shape is rotated is called the axis of revolution. For example, when you rotate a rectangle around a line that contains one of its sides, the solid of revolution that is produced is a cylinder.
Sketching and Describing Solids of Revolution
Sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid. a.^9
9
4 4
b.
2
5
SOLUTION
a.^9
4
b.
2
5
The solid is a cylinder with a height of 9 and a base radius of 4.
The solid is a cone with a height of 5 and a base radius of 2.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid.
7. 3
4
8 8
6
7
7
Section 11.4 Three-Dimensional Figures 621
11.4 Exercises Dynamic Solutions available at BigIdeasMath.com
In Exercises 3–6, match the polyhedron with its name.
3. 4.
A. triangular prism B. rectangular pyramid
C. hexagonal pyramid D. pentagonal prism
In Exercises 7–10, tell whether the solid is a polyhedron. If it is, name the polyhedron. (See Example 1.)
7. 8.
In Exercises 11 − 14, describe the cross section formed by the intersection of the plane and the solid. (See Example 2.)
11. 12.
In Exercises 15–18, sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid. (See Example 3.)
15.
8
8
8
8
6
6
3
3
2 2 5
5
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. VOCABULARY A(n) __________ is a solid that is bounded by polygons. 2. WHICH ONE DOESN’T BELONG? Which solid does not belong with the other three? Explain your reasoning.
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
