[PDF] - Surface Area Progressive Mathematics Initiative & This PDF Document

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Surface Area & Volume

www.njctl.org March 7, 2012

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Slide 4 / 219 Understanding 3-Dimensional Drawings

Return to Table of Contents

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X Y Y X

2-dimensional drawings use only the x and y axes

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X Y Length width

Y X Length width Y X Length w i d t h

SLIDE 2

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Y X Z height height Y X

3-dimensional drawings include the x, y and z-axis. The z-axis is the third dimension. The third dimension is the height of the figure

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Y X Z height height Y X

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Y X Z height height Y X

x

Y

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Y X Z height Y X X Y

r

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To give a figure more of a 3-dimensional, lines that are not visible from the angle the figure is being viewed are drawn as dashed line segments. These are called hidden lines.

Y X Z height height

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The 3-Dimensional Figures discussed in this unit are: Prisms Pyramids Cylinders Cones Spheres

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Prisms have 2 congruent polygon bases. The sides of a base are called base edges. The segments connecting corresponding vertices are lateral edges.

A B C X Y Z

In this diagram: There are 2 bases: ABC & XYZ. There are 6 base edges: AB, BC, AC, XY, YZ, & XZ. There are 3 lateral edges: AX, BY, & CZ. This prism has a total of 9 edges.

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The polygons that make up the surface of the figure are called faces. The bases are a type of face and are parallel and congruent to each other. The lateral edges are the sides of the lateral faces.

A B C X Y Z

In this diagram: There are 2 bases: ABC & XYZ. There are 6 base edges: AB, BC, AC, XY, YZ, & XZ. There are 3 lateral edges: AX, BY, & CZ. This prism has a total of 9 edges.

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1 A B C D E F G H I A B C D E F M N P Q R S

Choose all of the base edges.

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2 A B C D E F G H I A B C D E F M N P Q R S

Choose all of the lateral edges.

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3 A B C D E F G H I A B C D E F M N P Q R S

Choose all of the edges.

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4 Chooses all of the bases.

A

AFSM

B

FERS

C

EDQR

D

ABCDEF

E

CDQP

F

BCPN

G

MNPQRS

H

ABNM

A B C D E F M N P Q R S

SLIDE 4

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5 Chooses all of the lateral faces.

A

AFSM

B

FERS

C

EDQR

D

ABCDEF

E

CDQP

F

BCPN

G

MNPQRS

H

ABNM

A B C D E F M N P Q R S

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6 Chooses all of the faces.

A

AFSM

B

FERS

C

EDQR

D

ABCDEF

E

CDQP

F

BCPN

G

MNPQRS

H

ABNM

A B C D E F M N P Q R S

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A pyramid has 1 base and the lateral edges go to a single point. This point is called the vertex.

A M N P Q R S This pyramid has: 6 lateral edges, 6 base edges, 12 edges (total)

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A pyramid is faces that are polygons. 1 base and triangles that are the lateral faces.

A M N P Q R S This pyramid has: 6 lateral faces, 1 base, 7 faces (total)

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7 Choose all of the base edges.

A B C D E F

K L M N V Slide 24 / 219

8 Choose all of the base edges.

A B C D E F

K L M N V

SLIDE 5

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9 How many edges does the pyramid have?

K L M N V Slide 26 / 219

10 Choose all of the lateral faces.

A

KNV

B

NMV

C

KLMN

D

VML

E

KLV

K L M N V Slide 27 / 219

11 Choose all of the bases.

A

KNV

B

NMV

C

KLMN

D

VML

E

KLV

K L M N V Slide 28 / 219

12 How many faces does the pyramid have?

K L M N V Slide 29 / 219 Slide 30 / 219

A cone, like a pyramid, has one base which is a circle.

. N V

is the base of the cone. V is the vertex of the cone.

SLIDE 6

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A sphere is a 3-dimensional circle in that every point on the sphere is the same distance from the center.

. C Slide 32 / 219

13 Which shape has 2 bases?

A

Prism

B

Pyramid

C

Cylinder

D

Cone

E

Sphere

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14 Which shape has a vertex?

A

Prism

B

Pyramid

C

Cylinder

D

Cone

E

Sphere

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15 Which shape has more base edges than lateral edges?

A

Prism

B

Pyramid

C

Cylinder

D

Cone

E

Sphere

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16 Which shape has no vertices?

A

Prism

B

Pyramid

C

Cylinder

D

Cone

E

Sphere

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Surface v. Solid

Return to Table of Contents

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Surfaces and Solids are 3-dimensional figures.

A surface is the shell

f a figure.

A solid is a filled figure. In drawings, a solid is shaded so you cannot see through it.

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17 Is the following object a solid or a surface? A Brick

A

Solid

B

Surface

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18 Is the following object a solid or a surface? An empty shoe box

A

Solid

B

Surface

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19 Is the following object a solid or a surface? A new can of soda

A

Solid

B

Surface

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20 Is the following object a solid or a surface? An ice cream cone

A

Solid

B

Surface

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A cross-section is the locus of points of the intersection of a plane and a space figure.

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Think about as if the plane were a knife and you were cutting the shape, what would the cut look like?

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Cross-sections of a surface are a 2-dimensional figure. Cross-sections of a solid are a 2-dimensional figure and its interior.

(The top can be removed to see the cross section.)

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21 What is the locus of points (cross-section) of a cube and a plane perpendicular to the base and parallel to the non-intersecting sides?

A square B

rectangle

C

trapezoid

D

hexagon

E

rhombus

F

parallelogram

G

triangle

H

circle

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22 What is the locus of points of a cube and a plane that contains the diagonal of the base and is perpendicular to the base?

A

square

B

rectangle

C

trapezoid

D

hexagon

E

rhombus

F

parallelogram

G

triangle

H

circle

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23 What is the locus of points of a cube and a plane that contains the diagonal of the base but does not intersect the opposite base?

A

square

B

rectangle

C

trapezoid

D

hexagon

E

rhombus

F

parallelogram

G

triangle

H

circle

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24 What is the locus of points of a cube and a plane that intersects all of the faces?

A

square

B

rectangle

C

trapezoid

D

hexagon

E

rhombus

F

parallelogram

G

triangle

H

circle

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25 Choose all of the following that have cross-sections that consist of a 2-dimensional shape and its interior.

A

a brick

B

a balloon

C

an empty soda can

D

a stick of butter

E

a wrapping paper tube

F

a baseball

G

a straw

H

a book

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Right v. Oblique

Return to Table of Contents

Slide 51 / 219 Oblique lines- two lines that intersect and are not perpendicular. Slide 52 / 219

Right Triangular Prism Oblique Triangular Prism

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Right Cylinder Oblique Cylinder How can a right prism or cylinder be distinguished from an oblique figure?

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Right Hexagonal Pyramid Oblique Hexagonal Pyramid A right pyramid with a regular polygon for the base is called a regular pyramid. So this surface could also be called a regular hexagonal pyramid.

SLIDE 10

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Right Cone Oblique Cone How can a right pyramid or cone be distinguished from an oblique figure?

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Right Triangular Prism Oblique Triangular Prism What shape is each lateral face? What shape is each lateral face?

Slide 57 / 219 Naming a Figure

Right

r

Oblique Shape

f

Base Pyramid

r

Prism Solid

r

Surface Since cones and cylinders always circles as bases: Right

r

Oblique Solid

r

Surface Cylinder

r

Cone A sphere is neither right or oblique: Solid

r

Surface Spherical