lesson 9-2 prisms and cylinders

Lesson

Lesson 9-2

Prisms and Cylinders 525

Prisms and Cylinders9-2

Recall that polygons and polygonal regions are different. A polygon is the boundary of a polygonal region. The region is the union of the boundary and its interior.

A similar distinction is made in three dimensions, as shown by the above drawings. The person in the space at the right is in the interior of a surface consisting of six rectangular regions in six different planes. Roughly, a surface is a 3-dimensional fi gure that separates space into two regions. If one of the regions is bounded, all distances between points in the region are less than some fi xed number, then that region is the interior of the surface and the other region is the exterior of the surface. A solid is the union of a surface and its interior. For example, a balloon is a surface, while a bowling ball is a solid. The fi gure at the left above pictures a solid. When drawing, you can distinguish a solid from a surface by shading and showing no hidden lines.

Not all surfaces have interiors and exteriors. Some surfaces (like planes) separate space into two unbounded regions. But the surfaces you will study in this chapter are bounded.

Vocabulary

surface

interior of a surface

exterior of a surface

solid

face of a surface

edge of a surface

vertices of a surface

cylindrical solid

cylindrical surface

bases of a cylindrical solid

lateral surface

lateral face of a prism

height, altitude of a solid

right solid

oblique solid

cylinder

prism

lateral edge of a prism

regular prism

cube

a. F and G are two non-overlapping regions, each with an area of 19. What is the area of their union?

b. F and G are two regions, each with an area of 19. The area of their union is 30. Do they overlap?

c. In Part b, what is the area of the overlap between the two regions?

Mental Math

BIG IDEA Prisms and cylinders are two special types of cylindrical surfaces.

SMP_SEGEO_C09L02_525-531.indd 525SMP_SEGEO_C09L02_525-531.indd 525 5/21/08 12:28:30 PM5/21/08 12:28:30 PM

526 Three-Dimensional Figures

Chapter 9

Cylindrical Solids and Surfaces

The words cube, cylinder, and prism refer to types of 3-dimensional fi gures called cylindrical solids.

When a plane intersects a surface in a polygonal region, that region is called a face of the surface or face of the solid the surface determines. The sides of the polygonal region are called edges of the surface, and the vertices of the polygonal region are called vertices of the surface. For example, in the diagram at the right, M is a vertex,

____ MN is an edge, and rectangle

MNPQ is a face.

QY1

Cylindrical Solids and Surfaces

Examine the fi gures in the diagram above. You might notice that, while they are very different from one another, they also share a few important characteristics. For instance, in each fi gure, part of the surface consists of congruent regions in parallel planes. This suggests that all of these shapes can be created by choosing a fi gure and then translating it through space. This process is shown in the diagram below, where the regular pentagon at the left is translated by

� PQ to create the 3-dimensional fi gure at the right.

When a solid 3-dimensional shape is created using the method described above, the shape is called a cylindrical solid.

Defi nitions of Cylindrical Solid, Cylindrical Surface

A cylindrical solid is the set of points between a region and its translation image in space, including the region and its image. The boundary of a cylindrical solid is called a cylindrical surface.

In a cylindrical solid, the region that was translated and its image are its bases. In the fi gure above at the right, the bases are pentagons ABCDE and A′B′C ′D′E′. Bases are always congruent and are always in parallel planes.

QY1

Name another vertex, edge, and face that are shown in the diagram above.



Lesson 9-2

The rest of the surface of the solid is called the lateral surface. When the bases are polygons, the lateral surface is made up of lateral

faces, such as parallelogram A′E′EA in the fi gure on the previous page. The intersection of two lateral faces is called a lateral edge. The height or altitude of the solid is the distance between the planes of its bases. In the fi gure, the height is D′F. When the translation vector is perpendicular to the planes containing the bases, we say that the solid is a right solid; otherwise it is an oblique solid.

Example

The surface at the right is made up of six rectangles and two regular

hexagons. In the solid shown at the right, identify a base, a lateral

face, and a lateral edge. Finally, tell whether the solid is right or

oblique.

Solution Hexagon ABCDEF is a base. Rectangle AFPQ

is a lateral face.

___

DN is a lateral edge. The fi gure is right

because the lateral edges are perpendicular to the base.

Two types of cylindrical surfaces are particularly common and important.

Defi nitions of Cylinder, Prism

A cylinder is a cylindrical surface whose base is a circle.

A prism is a cylindrical surface whose base is a polygon.

While the base of a cylinder must be a circle, the base of a prism can be any polygonal region. We name a prism by the name of its base. For instance, if the base is a triangle, the prism is called a triangular prism. The faces of the lateral surface of a prism are always parallelograms. Many prisms that you will encounter will be right prisms whose bases are regular polygons. These are called regular prisms. A regular prism is a right prism whose bases are regular polygons. Thus, the shape in the Example is a regular hexagonal prism. If we wanted to indicate that we are including the interior of this fi gure, we would add the word solid to our description, so it would be a solid regular hexagonal prism. The prism above the defi nition of cylindrical solid on the previous page is a solid oblique pentagonal prism.

QY2

One kind of prism that you have seen before is a cube. A cube is a right square prism in which all 6 faces are congruent squares.

QY2

If a fi gure is a regular polygonal prism, describe the shapes that make up its lateral surface.



Chapter 9

Hierarchy of Cylindrical Surfaces

From their defi nitions, the various types of cylindrical surfaces fi t nicely into a hierarchy.

regular prism box

right prism oblique prism

cylindrical surface

cylinder prism

cube

right cylinder oblique cylinder

Activity

MATERIALS DGS-3D

Step 1 Use the following instructions to construct a pentagonal prism using a DGS-3D.a. Construct a point P on the plane.

b. Construct a line through P that is perpendicular to the plane.

c. Construct vector �

v on the perpendicular line.

d. Construct a regular pentagon on the plane. Your screen should look like the one at the right.

e. Construct a pentagonal prism using the pentagon and the vector.

Step 2 Copy the table below. Rotate the pentagonal prism in space and carefully count each element to complete the fi rst row of the table.

Step 3 Complete the table. If you need to use a DGS to construct the hexagonal and octagonal prisms, use the instructions from Step 1 but use a different shape in Part d. The last row of the table asks you to generalize your fi ndings for any prism in which the base has n sides.

Step 4 How many vertices, edges, and faces does a 100-gonal prism have?

Name Faces Vertices Edges

Pentagonal Prism ? ? ?

Hexagonal Prism ? ? ?

Octagonal Prism ? ? ?

� � � �

n-gonal Prism ? ? ?

Name Faces Vertices Edges

Pentagonal Prism ? ? ?

Hexagonal Prism ? ? ?

Octagonal Prism ? ? ?

� � � �

n-gonal Prism ? ? ?



Lesson 9-2

QY3

Questions

COVERING THE IDEAS

1. What is the difference between a surface and a solid?

2. In the fi gure below, �ABF is a translation image in space of �DCE.

F

AB

E

D C

a. What is the general name for the fi gure? b. Name the bases. c. What is the specifi c name for the fi gure? d. Name the lateral faces. e. List the edges.

In 3 and 4, refer to the regular hexagonal prism in the example.

3. Name the four pairs of opposite faces. 4. Name all edges that are parallel to

−−

EF .

5. True or False The height of a right prism equals the length of a lateral edge.

6. Multiple Choice Which is not true? A Every prism is a cylindrical surface. B Every cylinder is a cylindrical surface. C Every prism is a cylinder.

7. Explain why a cube is a regular prism.

8. The solid at the right was created by translating a regular decagonal region through space. Name this shape as specifi cally as possible.

QY3

If a prism has 30 vertices, how many sides does the base have?



Chapter 9

APPLYING THE MATHEMATICS

In 9−11, use the fact that the lateral edges of an oblique

cylinder or prism are not perpendicular to the planes of the

bases. Such cylindrical solids seem to lean. The amount of

lean can be measured from the perpendicular. For example,

the cylinder at the right has a 49º lean.

9. Sketch a cylinder that has a 20º lean. 10. Sketch a rectangular prism with a 70º lean. 11. How does the amount of lean compare with the measure of the

angle a lateral edge of the fi gure makes with the plane of a base?

12. The right cylinder pictured at the right has height 10 cm and base area 25π cm2. Find PQ.

13. The base of the regular triangular prism below has perimeter 12 units and the height of the prism is 16 units.

a. Find the area of one of its lateral faces. b. Find the area of one of its bases.

14. a. How many vertices does a prism have if its base is an octagon? b. What if its base is an 18-gon?

15. A restaurant uses drinking glasses that are right cylinders 7 inches high and have a diameter of 3 inches. They want their straws to always stick at least 2 inches out of the glass. What is the shortest straw they should buy?

16. Consider an oblique square prism. a. How many vertices does the prism have? b. How many faces are squares? c. What kind of shapes are the lateral faces?

17. True or False If the statement is true, explain why. If it is false, explain why or provide a counterexample.

a. All of the lateral edges of a prism are congruent. b. The lateral edges of a prism are perpendicular to the bases. c. The lateral faces of a prism are always congruent. d. A cube is the only prism in which all of the faces (including

bases and lateral faces) are congruent.

P

Q



Lesson 9-2

18. In the right triangular prism at the right, m∠NMP = 30, ____

MN ⊥ ___

NP , MP = 12, and NPOS is a square. Find MW.

19. A cube is outlined by sticks, so it is hollow and you can stick your hand through it. If one of the sticks has length s, what is the total length of the sticks needed to make the cube?

REVIEW

20. When lines � and p are both perpendicular to line m, each of the following is possible. Draw a convincing picture of each situation. (Lesson 9-1)

a. � || p b. � ⊥ p c. � and p are skew.

21. A basketball player dribbles a ball and it returns back up to his hand. If the path the ball travels is a straight line, what is true about this line and the plane described by the fl oor? (Lessons 9-1, 4-3)

22. In the fi gure at the right, O is the center of both circles. The inner circle has radius 2, the outer circle has radius 3, m � AC = m � BD = 90º, and CD = 1. Find the length of the longest path in the fi gure from A to B that never retraces itself. (Lesson 8-8)

23. Consider the rectangular region at the right. (Lesson 8-1)

a. If you had a pair of scissors, and made exactly one straight cut taking part of the fi gure away, could either fi gure have a greater perimeter than the rectangle?

b. Trace the rectangle, and draw where you could make two straight cuts (possibly not through the rectangle) so that the resulting fi gure would have a greater perimeter than the original rectangle.

24. Let A = (0, 0) and B = (1, 0). Let T be a translation with translation vector � v , T(A) = C, and T(B) = D. Suppose that the segments

___ AB and

____ CD intersect. (Lesson 4-6)

a. What can you say about the direction of � v ? b. What can you say about the magnitude of � v ?

EXPLORATION

25. Some solid prisms have special properties relative to light. What are these properties?

26. Find two nonmathematical meanings of the word lateral in a dictionary. How is the mathematical meaning of this word related to these nonmathematical meanings?

W

M

NP

O

S

B A

C

O

D

QY ANSWERS

1. Answers vary. Sample: P, ___PR , QPRS

2. congruent rectangles

3. 15


lesson 9-2 prisms and cylinders

Documents