Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron...

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Chapter 12 Chapter 12 Surface Area and Volume Surface Area and Volume

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Transcript of Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron...

Page 1: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Chapter 12Chapter 12Surface Area and VolumeSurface Area and Volume

Page 2: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Chapter 12 ObjectivesChapter 12 Objectives Define polyhedronDefine polyhedron Utilize Euler’s TheoremUtilize Euler’s Theorem Identify cross sectionsIdentify cross sections Identify a prism and cylinderIdentify a prism and cylinder Identify a pyramid and coneIdentify a pyramid and cone Calculate the surface area of special Calculate the surface area of special

polyhedrapolyhedra Calculate the volume of special polyhedraCalculate the volume of special polyhedra Identify a sphereIdentify a sphere Calculate the surface area and volume of a Calculate the surface area and volume of a

spheresphere Compare quantities between similar Compare quantities between similar

polyhedrapolyhedra

Page 3: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.1Lesson 12.1Exploring SolidsExploring Solids

Page 4: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.1 ObjectivesLesson 12.1 Objectives

Define polyhedronDefine polyhedron Use properties of polyhedraUse properties of polyhedra Utilize Euler’s TheoremUtilize Euler’s Theorem

Page 5: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

PolyhedronPolyhedron

A A polyhedronpolyhedron is a solid made of polygons. is a solid made of polygons. Remember, polygons are 2-D shapes with line Remember, polygons are 2-D shapes with line

segments for sides.segments for sides. The polygons form The polygons form facesfaces, or sides of the , or sides of the

solid.solid. An An edgeedge of a of a polyhedronpolyhedron is the line is the line

segment that is formed by the intersection segment that is formed by the intersection of 2 of 2 facesfaces.. Typically the sides of the polygon Typically the sides of the polygon facesfaces..

A A vertexvertex of a of a polyhedronpolyhedron is a point in is a point in which 3 or more which 3 or more edgesedges meet meet Typically the corners of the polygon Typically the corners of the polygon facesfaces..

Page 6: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example of a PolyhedronExample of a PolyhedronNotice that a polyhedron is nothing more than a 3-dimensional shape.

It has some form of a height, a width, and a depth.

The plural of polyhedron is polyhedra.

Faces

Edges

Vertices

Page 7: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 1Example 1

Determine if the following figures are polyhedra.Determine if the following figures are polyhedra.Explain your reasoning.Explain your reasoning.

No

There are no faces.There are no polygons.

Yes

All faces arepolygons.

No

One of the facesis not a polygon.

Page 8: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Regular vs ConvexRegular vs Convex

A A polyhedronpolyhedron is is regularregular if if ALLALL of of its its facesfaces are are regular polygons.regular polygons. Some polyhedron Some polyhedron

are are regularregular, some , some are not!are not!

A A polyhedronpolyhedron is is convexconvex if no line if no line segment segment connecting two connecting two interior points interior points leaves the leaves the polyhedronpolyhedron and re- and re-enters.enters.

No dents!No dents! If there are dents, If there are dents,

then the then the polyhedronpolyhedron is is concaveconcave..

Page 9: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Cross SectionsCross Sections

When you take a plane and cut When you take a plane and cut through a solid, the resulting shape of through a solid, the resulting shape of the surface is called the the surface is called the cross cross sectionsection.. When asked to identify a When asked to identify a cross sectioncross section, ,

you need to identify the polygon formed.you need to identify the polygon formed. The plane acts like a knife blade and cuts The plane acts like a knife blade and cuts

through the solid.through the solid.

Page 10: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Theorem 12.1:Theorem 12.1:Euler’s TheoremEuler’s Theorem

The number of The number of facesfaces ( (FF)), the number of , the number of verticesvertices ( (VV)), and the number of , and the number of edgesedges ( (EE) ) in a in a polyhedronpolyhedron are related by are related by FF + + VV = = EE + 2 + 2

This is commonly used to find one of the variables above.This is commonly used to find one of the variables above. Typically this is because it is hard to see all the faces, Typically this is because it is hard to see all the faces,

vertices, and edges.vertices, and edges.

F = 6

V = 8

E = ???

6 + 8 = E + 2

14 = E + 2

12 = E

Page 11: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 2Example 2

Find the number of vertices, faces, and edges each polyhedron has.Find the number of vertices, faces, and edges each polyhedron has.

FF + + VV = = EE + 2 + 2

FF + + 88 = = 1212 + 2 + 2

FF + + 88 = = 1414

FF = = 66

FF = 5 = 5

VV = 6 = 6

55 + + 66 = = E E + 2+ 2

11 = 11 = E E + 2+ 2

E E = 9= 9

FF = 6 = 6

VV = 6 = 6

66 + + 66 = = E E + 2+ 2

12 = 12 = E E + 2+ 2

E E = 10= 10

FF = 8 = 8

VV = 12 = 12

88 + + 1212 = = E E + 2+ 2

20 = 20 = E E + 2+ 2

E E = 18= 18

You do not have to use the formula to find edges every time. You can use it to find any of the missing quantities as long as you know the other 2, if you use the formula at all!

Page 12: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Regular PolyhedraRegular Polyhedra

There are only five regular polyhedra, There are only five regular polyhedra, called called Platonic solidsPlatonic solids..

Named after Greek mathematician and Named after Greek mathematician and philosopher Platophilosopher Plato

They use regular triangles, squares, They use regular triangles, squares, and regular pentagons to form solids.and regular pentagons to form solids. Those are the only 3 shapes that can be Those are the only 3 shapes that can be

used.used. That is because when their vertices are That is because when their vertices are

butted together, their interior angles can add butted together, their interior angles can add to 360to 360oo..

Page 13: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Platonic SolidsPlatonic Solids

NameName FaceFace ShapeShape

##

FacesFaces##

VerticesVertices##

EdgesEdges

TetrahedronTetrahedron TriangleTriangle 44 44 66

CubeCube SquareSquare 66 88 1212

OctahedronOctahedron TriangleTriangle 88 66 1212

DodecahedroDodecahedronn

PentagonPentagon 1212 2020 3030

IcosahedronIcosahedron TriangleTriangle 2020 1212 3030

Page 14: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Homework 12.1Homework 12.1

In ClassIn Class 1-91-9

p723-726 p723-726

HWHW 10-55, 60-70 ev10-55, 60-70 ev

Due TomorrowDue Tomorrow

Page 15: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.2Lesson 12.2Surface Area of Prisms and CylindersSurface Area of Prisms and Cylinders

Page 16: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.2 ObjectivesLesson 12.2 Objectives

Identify a prismIdentify a prism Calculate the surface area of a Calculate the surface area of a

prismprism Identify a cylinderIdentify a cylinder Calculate the surface area of a Calculate the surface area of a

cylindercylinder Construct a two-dimensional net for Construct a two-dimensional net for

three-dimensional solidsthree-dimensional solids

Page 17: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

PrismsPrisms

A A prismprism is a polyhedron with two is a polyhedron with two congruentcongruent facesfaces that are that are parallelparallel to each other. to each other.

The The congruentcongruent facesfaces are called are called basesbases.. The The basesbases must be must be parallelparallel to each other. to each other.

The other The other facesfaces are called are called lateral faceslateral faces.. These are always rectangles or parallelograms or These are always rectangles or parallelograms or

squares.squares. When naming a When naming a prismprism, they are always named , they are always named

by the shape of their by the shape of their basesbases..

Triangular Prism

Page 18: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Parts of a PrismParts of a Prism The The edgesedges that connect opposing that connect opposing basesbases are are

called called lateral edgeslateral edges.. The The heightheight of a prism is the of a prism is the perpendicularperpendicular

distance between the distance between the basesbases.. In a In a right prismright prism, the length of the , the length of the lateral edgelateral edge is is

the the heightheight.. A A right prismright prism is one that stands up straight with the is one that stands up straight with the

lateral edgeslateral edges perpendicular to the perpendicular to the basesbases.. In an In an oblique prismoblique prism, the , the heightheight must be drawn in must be drawn in

so that it is perpendicular to both so that it is perpendicular to both basesbases.. An An oblique prismoblique prism is one that is slanted to one side or the is one that is slanted to one side or the

other.other. The length of the slanted The length of the slanted lateral edgelateral edge is called the is called the slant slant

heightheight..

Page 19: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Examples of PrismsExamples of Prisms

slant h

eig

ht

lateral edge

base

lateral face

Page 20: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 3Example 3

Answer the following questions:Answer the following questions:

a)a) What kind of figure is the base?What kind of figure is the base?a)a) HexagonHexagon

b)b) What kind of figure is each What kind of figure is each lateral face?lateral face?

b)b) RectangleRectangle

c)c) How many lateral faces does How many lateral faces does the figure have?the figure have?

c)c) 66i.i. Same number as the sides of the Same number as the sides of the

base!base!

d)d) Give the mathematical name of Give the mathematical name of the geometrical solid.the geometrical solid.

d)d) Hexagonal PrismHexagonal Prism

Page 21: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

AreaArea

The The surface areasurface area of a of a prismprism is the is the sum of the areas of all the faces sum of the areas of all the faces and bases.and bases.

The The lateral arealateral area of a of a prismprism is the is the sum of the areas of the sum of the areas of the lateral lateral facesfaces..

Page 22: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Theorem 12.2:Theorem 12.2:Surface Area of a PrismSurface Area of a Prism

The The surface areasurface area ( (SS)) of a of a right right prismprism can be found using the can be found using the formulaformula SS = 2 = 2BB + P + Phh

BB is is areaarea of the of the basebase A = ½ aP A = ½ aP (any polygon)(any polygon) A = lw A = lw (rectangle)(rectangle) A = bh A = bh (parallelogram)(parallelogram) A = ½ bh A = ½ bh (triangle)(triangle) A = sA = s22 (square)(square)

P is perimeter of the P is perimeter of the basebase hh is is heightheight of the prism of the prism

Page 23: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 4Example 4

Find the surface area of the following:Find the surface area of the following:

SS = 2 = 2BB + P + Phh

SS = 2 = 2(5(5•6)•6) + P + Phh

SS = 2 = 2(5(5•6)•6) + (5+6+5+6) + (5+6+5+6)hh

SS = 2 = 2(5(5•6)•6) + (5+6+5+6) + (5+6+5+6)(7)(7)

SS = 2 = 2(30(30)) + (22) + (22)(7)(7)

SS = 60 + 154 = 60 + 154

SS = 214 m = 214 m22 areaarea

SS = 2 = 2BB + P + Phh

SS = 2 = 2((11//22••1212•5)•5) + P + Phh

SS = 2 = 2(5(5•6)•6) + (5+12+ ) + (5+12+ )hh

SS = 2 = 2(5(5•6)•6) + (30) + (30)hh

SS = 2 = 2(30(30)) + (30) + (30)(10)(10)

SS = 60 + 300 = 60 + 300

SS = 360 m = 360 m22

?

a2 + b2 = c2

52 + 122 = c2

52 + 122 = c2

25 + 144 = c2

25 + 144 = c2

169 = c2

c = 13

1313

Page 24: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

CylinderCylinder

A A cylindercylinder is a solid with is a solid with congruentcongruent and and parallelparallel circlescircles for for basesbases..

height

Page 25: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Theorem 12.3:Theorem 12.3:Surface Area of a CylinderSurface Area of a Cylinder

The The surface areasurface area ( (SS) ) of a of a right right cylindercylinder is is SS = 2 = 2BB + + CChh

BB is is areaarea of the of the basebase A = A = r r22

CC is is circumferencecircumference of the of the basebase C = 2C = 2 r r oror C = C = d d

hh is is heightheight of the cylinder of the cylinder

Page 26: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 5Example 5Find the surface area of the following cylinder.Find the surface area of the following cylinder.Round your answer to the nearest hundreth.Round your answer to the nearest hundreth.

SS = 2 = 2BB + + CChh

SS = 2 = 2rr22 + + CChh

SS = 2 = 2rr22 + + 22rrhh

SS = 2 = 2(9)(9)22 + + 22(9)(9)(7)(7)

SS = 2 = 2(81)(81) + 2 + 2(63)(63)

SS = 162 = 162 + 126 + 126

SS = 288 = 288

SS = 508.94 + 395.84 = 508.94 + 395.84= 904.78= 904.78ftft22

Page 27: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

NetsNets

A A netnet is a two-dimensional drawing of a is a two-dimensional drawing of a three-dimensional solid.three-dimensional solid. If you were to unfold a solid, the If you were to unfold a solid, the netnet would would

show what it looks like.show what it looks like. Every solid has a Every solid has a netnet..

However, there are only certain ways to draw However, there are only certain ways to draw a a netnet for each solid.for each solid.

Page 28: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 6Example 6

Identify the solid formed by the given net.Identify the solid formed by the given net.Remember: Solids have a full name (2 parts)Remember: Solids have a full name (2 parts)

Triangular Prism Square Prism

a.k.a Cube

Cylinder

Page 29: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Homework 12.2Homework 12.2

In ClassIn Class 1,3-121,3-12

p730-p730-734734

HWHW 13-37, 43, 44, 50-60 ev13-37, 43, 44, 50-60 ev

Due TomorrowDue Tomorrow

Page 30: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.3Lesson 12.3Surface Area of Pyramids and ConesSurface Area of Pyramids and Cones

Page 31: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.3 ObjectivesLesson 12.3 Objectives

Identify a pyramidIdentify a pyramid Calculate slant heightCalculate slant height Calculate the surface area of a Calculate the surface area of a

pyramidpyramid Identify a coneIdentify a cone Calculate the surface area of a coneCalculate the surface area of a cone

Page 32: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

PyramidPyramid

A A pyramidpyramid is a polyhedron with is a polyhedron with oneone basebase and and lateral faceslateral faces that meet at that meet at one common vertex.one common vertex. The The basebase must be a polygon. must be a polygon.

Not necessarily a square!Not necessarily a square! The The lateral faceslateral faces will always be triangles. will always be triangles.

Name the Name the pyramidpyramid by its by its basebase shape. shape.

Square

PyramidPentagonal

Pyramid

Page 33: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Regular PyramidsRegular Pyramids A A regular pyramidregular pyramid has a has a regularregular polygon for a polygon for a

basebase, and the common , and the common vertexvertex is directly above is directly above the center of the the center of the basebase..

The The heightheight of a of a pyramidpyramid is the is the perpendicularperpendicular distance from the distance from the basebase to the common to the common vertexvertex.. The The heightheight of a of a regular pyramidregular pyramid is the length of the line is the length of the line

drawn from the center of the drawn from the center of the basebase straight up to the straight up to the common common vertexvertex..

The The slant heightslant height only exists in only exists in regular pyramidsregular pyramids and is and is the length of a line drawn from the the length of a line drawn from the basebase up the up the lateral lateral faceface to the common to the common vertexvertex..

Base

Slant heightHeight

Lateral edge

Page 34: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Finding Slant HeightFinding Slant Height

To find the To find the slant heightslant height, you must know or be , you must know or be able to calculateable to calculate heightheight of the pyramid of the pyramid apothemapothem of the of the basebase of the pyramid of the pyramid

can be found knowing one side of the can be found knowing one side of the basebase..

The reason you need those quantities is The reason you need those quantities is because they form a because they form a hiddenhidden right triangle. right triangle. Then Pythagorean Theorem can be used to find the Then Pythagorean Theorem can be used to find the

missing missing slant heightslant height..

Slant height

Height

Apothem

Page 35: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 7Example 7Find the slant height for the following pyramid.Find the slant height for the following pyramid.

Remember, you must make a right triangle using Remember, you must make a right triangle using heightheight, , slant heightslant height, and , and apothemapothem..

10

24x

c2 = a2 + b2

x2 = 102 + 242

x2 = 100 + 576

x2 = 676

x = √676

x = 26 cm

Page 36: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Theorem 12.4:Theorem 12.4:Surface Area of a Regular Surface Area of a Regular

PyramidPyramid The The surface area (S)surface area (S) of a regular of a regular

pyramid ispyramid is SS = = BB + ½P + ½Pll

BB is the is the basebase P is the perimeter of the P is the perimeter of the basebase l l is theis the slant heightslant height

Page 37: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 8Example 8

Find the surface area of the following pyramid.Find the surface area of the following pyramid.

SS = = BB + ½P + ½Pll

SS = = (9(9•9)•9) + ½P + ½Pll

SS = = (9(9•9)•9) + ½(9+9+9+9) + ½(9+9+9+9)ll

SS = = (9(9•9)•9) + ½(9+9+9+9) + ½(9+9+9+9)(10)(10)

SS = = (81(81)) + ½(36) + ½(36)(10)(10)

SS = = (81(81)) + ½(360) + ½(360)

SS = = (81(81)) + 180 + 180

SS = 261 m = 261 m22 areaarea

Page 38: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

ConeCone

A A conecone has a circular has a circular basebase and a and a vertexvertex that is not in the same plane that is not in the same plane as the as the basebase.. All the same rules apply with a All the same rules apply with a conecone as as

they do with a they do with a pyramidpyramid..

height

B

B

Page 39: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Theorem 12.5:Theorem 12.5:Surface Area of a ConeSurface Area of a Cone

The The surface areasurface area of a right of a right conecone is is SS = = rr22 + + rrll

r is radius of the r is radius of the basebase circle circle l l is the is the slant heightslant height

Page 40: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Homework 12.3Homework 12.3

In ClassIn Class 1-131-13

p738–p738–741741

HWHW 14-39,44-46,50-5314-39,44-46,50-53

Due TomorrowDue Tomorrow

Page 41: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.4Lesson 12.4Volume of Prisms and Volume of Prisms and CylindersCylinders

Page 42: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.4 ObjectivesLesson 12.4 Objectives

Utilize the Volume PostulatesUtilize the Volume Postulates Calculate the volume of a prismCalculate the volume of a prism Calculate the volume of a cylinderCalculate the volume of a cylinder Apply Cavalieri’s PrincipleApply Cavalieri’s Principle

Page 43: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

VolumeVolume

The The volumevolume of any solid is the amount of any solid is the amount of space contained in its interior.of space contained in its interior.

The The volumevolume is measured in cubed units is measured in cubed units mm33

cmcm33

ftft33

inin33

unitsunits33

Page 44: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Volume PostulatesVolume PostulatesPostulate 27: Volume of a CubePostulate 27: Volume of a Cube

The The volumevolume of a cube is the cube of of a cube is the cube of the length of its side.the length of its side. V = sV = s33

4 in

V = (4 in)3

V = 64 in3

Page 45: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Volume PostulatesVolume Postulates

Postulate 28:Postulate 28:Volume Volume CongruenceCongruence

If two polyhedra are If two polyhedra are congruentcongruent, then they , then they have the same volume.have the same volume.

Postulate 29:Postulate 29:Volume AdditionVolume Addition

The volume of a solid is The volume of a solid is the sum of the volume of the sum of the volume of all its all its nonoverlappingnonoverlapping parts.parts.

Page 46: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Theorem 12.6:Theorem 12.6:Cavalieri’s PrincipleCavalieri’s Principle

If two solids have the same If two solids have the same heightheight and the same and the same cross-sectionalcross-sectional area area at at everyevery level, then they have the level, then they have the same volume.same volume. So whether the solid is tilted or So whether the solid is tilted or

straight up, the volume is the same as straight up, the volume is the same as long as the base area is the same size long as the base area is the same size all the way up the solid.all the way up the solid.

B

BB

heightheight

Page 47: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Volume TheoremsVolume Theorems

Theorem 12.7:Theorem 12.7:Volume of a PrismVolume of a Prism

The The volumevolume ((VV))of a of a prismprism is is VV = = BBhh

VV is is volumevolume BB is area of the is area of the basebase hh is the is the heightheight of the of the prismprism

Theorem 12.8:Theorem 12.8:Volume of a CylinderVolume of a Cylinder

The The volumevolume ( (VV)) of a cylinder of a cylinder isis VV = = BBhh

BB is area of the circular is area of the circular basebase rr22

B

height

B

Page 48: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 9Example 9

Find the volume of the following figures.Find the volume of the following figures.

VV = = BBhh

VV = = ((rr2 2 ))hh

VV = = (((7)(7)2 2 ))hh

VV = = (((7)(7)2 2 ))(2)(2)

VV = = (49(49))(2)(2)

VV = 98 = 98 = 307.88 cm= 307.88 cm33

VV = = BBhh

VV = = [[1/2a(ns)]]hhVV = = [[1/2(2.8)(ns)]]hh

VV = = [[1/2(2.8)(5)(s)]]hh

VV = = [[1/2(2.8)(5)(4)]]hh

VV = = [[1/2(2.8)(5)(4)]](6)(6)

VV = = [[28]](6)(6)

VV = 168 m = 168 m33

Page 49: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Homework 12.4Homework 12.4

In ClassIn Class 1-91-9

p746-p746-749749

HWHW 10-37, 48, 49, 52-60 ev10-37, 48, 49, 52-60 ev

Due TomorrowDue Tomorrow

Page 50: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.5Lesson 12.5Volume of Pyramids and Volume of Pyramids and ConesCones

Page 51: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.5 ObjectivesLesson 12.5 Objectives

Calculate the volume of a pyramidCalculate the volume of a pyramid Calculate the volume of a coneCalculate the volume of a cone

Page 52: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

The Ghost of CavalieriThe Ghost of Cavalieri

Cavalieri’s PrincipleCavalieri’s Principle is stated for is stated for ALLALL solids, which include solids, which include pyramidspyramids and and conescones

So the So the volumevolume of a of a pyramidpyramid or or conecone is is the same as long as the area of the the same as long as the area of the basebase is congruent and the solids have is congruent and the solids have the same the same heightheight It does not matter if the solid is slanted or It does not matter if the solid is slanted or

not, it only matters with overall not, it only matters with overall heightheight and area of the and area of the basebase..

Page 53: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Volume TheoremsVolume Theorems

Theorem 12.9:Theorem 12.9:Volume of a PyramidVolume of a Pyramid

The The volumevolume ( (VV)) of a of a pyramidpyramid isis VV = = 11//33BBhh

VV is is volumevolume BB is area of the is area of the basebase hh is is heightheight of the of the pyramidpyramid

Theorem 12.10:Theorem 12.10:Volume of a ConeVolume of a Cone

The The volumevolume ( (VV)) of a of a conecone is is VV = = 11//33BBhh

BB is area of the circular is area of the circular basebase

rr22

height

BB

Page 54: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 10Example 10

Find the volume of the following solids.Find the volume of the following solids.

VV = = 11//33BBhh

VV = = 11//33[[11//22bh]bh]hh

VV = = 11//33[[11//22(4)h](4)h]hh

VV = = 11//33[[11//22(4)(6)](4)(6)]hh

VV = = 11//33[[11//22(4)(6)](4)(6)](5)(5)

VV = = 11//33[12][12](5)(5)

VV = = 11//33(60)(60) = 20 m= 20 m33

VV = = 11//33BBhh

VV = = 11//33[[rr22]]hh

VV = = 11//33[[(3)(3)22]]hh

VV = = 11//33[[(3)(3)22]](8)(8)

VV = = 11//33[9[9]](8)(8)

VV = = 11//33(72(72))

VV = 24 = 24= 75.40 mm= 75.40 mm33

Page 55: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Homework 12.5Homework 12.5

In ClassIn Class 1-71-7

p755-p755-758758

HWHW 8-25, 30-34, 40-50 ev8-25, 30-34, 40-50 ev

Due TomorrowDue Tomorrow

Page 56: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.6Lesson 12.6Surface Area and Volume of Surface Area and Volume of SpheresSpheres

Page 57: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.6 ObjectivesLesson 12.6 Objectives

Define a sphereDefine a sphere Calculate the surface area of a Calculate the surface area of a

spheresphere Calculate the volume of a sphereCalculate the volume of a sphere

Page 58: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

SphereSphere

A A spheresphere is a set of points in space is a set of points in space that are equidistant from one given that are equidistant from one given point.point. A A spheresphere is a shell of points that are the is a shell of points that are the

same distance from the center.same distance from the center. A A spheresphere is a 3-dimensional circle. is a 3-dimensional circle.

Page 59: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Parts of a SphereParts of a Sphere The point inside the The point inside the spheresphere where all where all

points are equidistant to is called the points are equidistant to is called the center of the spherecenter of the sphere..

A A radius of the sphereradius of the sphere is a segment is a segment drawn from the drawn from the centercenter to a point on to a point on the the spheresphere..

A A chord of a spherechord of a sphere is a segment is a segment that joins any two points on the that joins any two points on the spheresphere.. The The diameterdiameter is also a is also a chordchord..

Any 2-dimensional circle that contains Any 2-dimensional circle that contains the the center of the spherecenter of the sphere is called a is called a great circlegreat circle.. The equator would be a The equator would be a great circlegreat circle..

Every Every great circlegreat circle of a of a spheresphere splits a splits a sphere into two congruent halves sphere into two congruent halves called called hemisphereshemispheres..

Page 60: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 11Example 11

Identify the following characteristics:Identify the following characteristics:

a)a) Name the center of the sphere.Name the center of the sphere.a)a) TT

b)b) Name a segment that is the radius of Name a segment that is the radius of the sphere.the sphere.

b)b) segment TSsegment TSsegment TQsegment TQsegment TPsegment TP

c)c) Name a chord of the sphere.Name a chord of the sphere.c)c) segment QRsegment QR

segment PSsegment PS

d)d) Find the circumference of the great Find the circumference of the great circle.circle.Write your final answers in terms of Write your final answers in terms of ..

d)d) 1414 m m

Page 61: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Theorem 12.11:Theorem 12.11:Surface Area of a SphereSurface Area of a Sphere

The The surface areasurface area ( (SS)) of a of a spheresphere is is SS = 4 = 4rr22

Page 62: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Theorem 12.12:Theorem 12.12:Volume of a SphereVolume of a Sphere

The The volumevolume ( (VV)) of a of a spheresphere is is VV = = 44//33rr33

Page 63: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Example 12Example 12Find the surface area and volume of the following sphere.Find the surface area and volume of the following sphere.

Round answer to the nearest hundredth.Round answer to the nearest hundredth.

SS = 4 = 4rr22

SS = 4 = 4(19)(19)22

SS = 4 = 4(361)(361)

SS = 1444 = 1444

SS = 4536.46 ft = 4536.46 ft22

VV = = 44//33rr33

VV = = 44//33(19)(19)33

VV = = 44//33(6859)(6859)

VV = ( = (27,43627,436//33))

VV = 28,730.91 ft = 28,730.91 ft33

Page 64: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Homework 12.6Homework 12.6

In ClassIn Class 1-91-9

p762-p762-765765

HWHW 10-28 ev, 45-48, 54-5710-28 ev, 45-48, 54-57

Due TomorrowDue Tomorrow

Page 65: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.7Lesson 12.7Similar SolidsSimilar Solids

Page 66: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Lesson 12.7 ObjectivesLesson 12.7 Objectives

Identify similar solidsIdentify similar solids Find and use the scale factor of Find and use the scale factor of

similar solidssimilar solids

Page 67: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Similar SolidsSimilar Solids Two solids with Two solids with equal ratiosequal ratios of corresponding of corresponding

linear measures are said to be linear measures are said to be similar solidssimilar solids.. That means that the two That means that the two similar solidssimilar solids must have must have

the same scale factor ratio for:the same scale factor ratio for: corresponding heightscorresponding heights corresponding radiicorresponding radii corresponding side lengthscorresponding side lengths

Page 68: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

ExampleExample

Determine if the solids are similar.Determine if the solids are similar.If so, find the scale factorIf so, find the scale factor..

Not similar

Because not all scale factors are equal

1:3

1:4

1:4

Similar

All spheres are similar!

Similar

Scale factor is 3:2

[--4--]

Similar

Scale factor is 5:2

Page 69: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Theorem 12:13Theorem 12:13Similar Solids TheoremSimilar Solids Theorem

If two solids have a scale factor of If two solids have a scale factor of a:ba:b, then the corresponding areas , then the corresponding areas have a ratio of:have a ratio of:

aa2 2 : b: b22

And the corresponding volumes And the corresponding volumes have a ratio of:have a ratio of:

aa3 3 : b: b33

Page 70: Chapter 12 Surface Area and Volume. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify.

Homework 12.7Homework 12.7

In ClassIn Class 1-81-8

pp769-772 769-772

HWHW 9-36, 40-48 ev9-36, 40-48 ev

Due TomorrowDue Tomorrow


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FeatureLesson Geometry Lesson Main 1. Draw a net for the figure. Use Euler’s Formula to solve. 2. A polyhedron with 12 vertices and 30 edges has how many.

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