Lesson 12.1 Exploring Solids

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Lesson 12.1Exploring SolidsToday, we will learn to> use vocabulary associated with solids> use Eulers Formula

Polyhedron ~ a solid formed by polygons

Prismsare polyhedra named by their basesname the base

Prisms have 2 parallel bases, Pyramids have 1 base.

Cylinders, Cones, & Spheres with curved surfaces are NOT polyhedra

Parts of Solids FacesEdgesVertexBases vs Lateral FacesHeight

F + V = E = + = 681412

F + V = E = + = 55108

Eulers Theorem# Faces + # Vertices = # Edges + 2F + V = E + 2

1. In a polyhedron,F = 8 V = 10 E = ?find the number of edges. F + V = E + 28 + 10 = E + 218 = E + 2___ edges16

2. In a polyhedron,F = 5 V = ? E = 8find the number of vertices. F + V = E + 25 + V = 8 + 25 + V = 10___ vertices5

3. In a polyhedron,F = ? V = 6 E = 9find the number of faces. F + V = E + 2F + 6 = 9 + 2F + 6 = 11___ faces5

A soccer ball is a polyhedron with 32 faces (20 hexagons & 12 pentagons). How many vertices does this polyhedron have?32 + V = 90 + 2# Edges =hexagonspentagons20(6)12(5) (120 + 60)= 90V = 60 vertices= 120= 60

A polyhedron can be convex or concave.

4.Describe the cross section shown.square

5.Describe the cross section shown.pentagon

6.Describe the cross section shown.triangle

A polyhedron is regular if all of its faces are congruent regular polygons.Platonic Solids

Lesson 11.1Angle Measures in PolygonsToday, we are going to> find angle measures in polygons

1802346n236054072011080(n2)1806090108120135(n2)180n

Theorem 11.1Polygon Interior Angles Theorem
The sum of the measures of the interior angles is____________(n  2)180

The measure of one interior angle in a regular polygon isCorollary to Theorem 11.1

1. Find the sum of the measures of the interior angles of a 30gon.(30  2) 180(28) 1805040

2.How many sides does a polygon have if the sum of the interior angles is 3240?n = 20 sides180(n2) = 3240n 2 = 18

3. What is the measure of each interior angle of a regular nonagon?140

4. Find x.180(7  2)180(5)900How many angles?7900 given angles = x900 783 = 117Sum?

5. Find x.180(5  2)180(3)540How many angles?5sum of all angles = 54043x 19 = 54013Sum?x =

12345Exploring Exterior AnglesGSP

Theorem 11.2Polygon Exterior Angles Theorem
The sum of the measures of one set of exterior angles in any polygon is_________360

The measure of one exterior angle in any regular polygon is

The number of sides in any regular polygon is# sides =

One exterior angle and its interior angle are always________________.supplementary

6. What is the measure of each exterior angle in a regular decagon? 36

7.How many sides does a regular polygon have if each exterior angle measures 40?= 9 sides

8.How many sides does a regular polygon have if each interior angle measures 165.6?Dont write this down, yet.180(n 2) = 165.6 n180n 360 = 165.6 n 360 = 14.4 nn =25 sides180 165.6 =First, find the measure of an exterior angle.= 25 sides14.4

9. Find x.3x + 90 + 70 + 80 + 60 =360x = 20 ??9070

Whats the measure of each interior angle of a regular pentagon?Whats the measure of each interior angle of a regular hexagon?= 108= 120A soccer ball is made up of 20 hexagons and 12 pentagons.

Lesson 11.2Perimeter & Areas of Regular PolygonsToday, we are going to> find the perimeter and area of regular polygons

101010 1. Find the area of this equilateral triangle6010 43.3

242424 2. Find the area of this equilateral triangle 249.4

sssEquilateral TrianglesArea =

A regular polygons area can be covered with isosceles triangles.

sideArea =Perimeter = side apothem side # sides # sidesside

Area of Regular Polygons
A = (side)(apothem)(# sides)
A = sa n

Perimeter of Regular Polygons
P = (side)(# sides)
P = s n

Find the area of the polygon.
3.a pentagon with an apothem of 0.8 cm and side length of 1.2 cmA = (1.2) (0.8) (5)A = 2.4 cm2A = (s) (a) (n)

Find the area of the polygon.
4.a polygon with perimeter 120 m and apothem 1.7 mA = (1.7) (120)A = 102 m2A = (s) (a) (n)

726045363022.55. Find the central angle of the polygon.6. Find the measure of this angle

The measure of the angle formed by the apothem and a radius of a polygon is3602(n)

5 cm10 cm7. Find the length of the apothem if the side length is 10 cm.3060

10 cm8. Find the area of the polygon.259.8 cm2

105a30s=a=n=106A = san= 259.8

126a36s=a=n=125A = san8.3= 249

12xa22.5s=a=n=9.28A = san11.1= 40812

14. Find the area of the square.6xs =45a

WorksheetPractice Problems

Lesson 11.3Perimeters and Areas of Similar FiguresToday, we are going to> explore the perimeters and areas of similar figures

1. Find the ratio of their Sides Perimeters Areas3 : 23 : 29 : 436 :54 :2424

2. Find the ratio of their Sides Perimeters Areas3 : 43 : 49 : 1660 :126 :80224

Theorem 11.5
In similar polygons Sides Perimeters Areas 3 : 2 3 : 2 9 : 43 : 4 3 : 4 9 : 16Do you notice a pattern? a : ba2 : b2a : b

PV = 63. The ratio of the area of PVQ to the area of RVT is 9:25. If the length of RV is 10 and the two triangles are similar, find PV.What is the ratio of their sides? x 10

4. The ratio of the sides of two similar polygons is 4 : 7. If the area of the smaller polygon is 36 cm2, find the area of the larger polygon.110.25 cm2What is the ratio of their areas? 36 x

Lesson 12.7Similar Solids

1. Find the ratio of sides.2:3Are they similar solids?3612

2. Find the ratio of their surface areas.112 : 252 reduces to 4:9 112 units2 252 units2

3. Find the ratio of their volumes.64 : 216 reduces to 8 : 27 64 units3216 units3

Theorem 12.13a2 : b2a3 : b3Ratio ofsidesRatio ofSurface AreaRatio ofVolumea : b2 : 34 : 98 : 27

4.1 : 161 : 64Scale FactorSAV1 : 42 : 54 : 258 : 1259 : 1003 : 1027 : 1000

5.A right cylinder with a surface area of 48 square centimeters and a volume of 45 cubic centimeters is similar to another larger cylinder. Their scale factor is 2:3. Find the surface area and volume of the larger solid.ratio of surface areas? 4:94948x4x = 9(48)x = 108 339.3

5.A right cylinder with a surface area of 48 square centimeters and a volume of 45 cubic centimeters is similar to another larger cylinder. Their scale factor is 2:3. Find the surface area and volume of the larger solid.ratio of volumes? 8:2782745x8x = 27(45)x = 151.875 477.1

Lesson 12.2, 12.3, 12.6Surface AreaToday, we will learn to> find the Surface Area of prisms, cylinders, pyramids, cones, and spheres

Why would we need to find surface area?

Surface Area of a Right Prism S = 2B + PH

S = 749282( ) + ( )( ) =294cm2Shape of the base?square

S = 2( ) + ( )( ) = 91615174m2Shape of the base?rectangle

2( )+( )( ) =S = 54363216 cm2Shape of the base?triangle

Surface Area of a Cylinder S = 2B + PHS = 2 r 2 + 2 r H

S = 2 ( ) 2 + 2 ( )( )4 251.3cm2Shape of the base?circler = 4 4 6

88

Surface Area of a Regular Pyramid S = B + P

10.86432( ) + ( )( ) = 236.8m2S = = 10.8

Surface Area of a ConeS = B + P

17628.3in2r = 8S = ( ) 2 + ( )( )8 8 17

Surface Area of a SphereS =4 r2

8 inr = S=
84(64)804.2in2

Each layer of this cake is 3 inches high. One can of frosting will cover 130 square inches of cake, how many cans do we need?S =641.34 in2B = san33 355We need 5 cans

Find the surface area.72 cm22( )+( )( ) =S = 4 88

Find the surface area.48 in2x2 = 32 + 42xx = 53 + 4 + 52( )+( )( ) =S = 6123

Find the surface area.S =2( )2 + 2( )( )967.61 in27715

Find the surface area.S =( ) + ( )( )144 in24ll 2 = 32 + 42 l = 5 64325

Find the surface area.S =63.33 m2( ) + ( )( )10.815710.8B?

Find the surface area.103.67 in2S = ( ) 2 + ( )( )3 3 8

Find the surface area.282.74 cm2x 2 = 52 + 122 x = 13 S = ( ) 2 + ( )( )5 5 13

Find the surface area.S =4( )2452.39 cm26

Lesson 12.4,12.5,12.6VolumeToday, we will learn to> find the Volume of prisms, cylinders, pyramids, cones, and spheres

Volume?5 times 3 is 15 4 layers of 15 60 cubes 5(3)(4) ? Why do we need to find volume?

Theorem 12.6Cavaleiris PrincipleIf 2 solids have the same height and the same crosssectional area at every level, then they have the same volume.

Rectangular Prism V = LWHVolume of a Triangular Prism V = bhHCube V = s3

V = ( )3=343 cm37

V = ( )( )( )=135 m33 5 9

V = ( )( )( )=162 cm312 9 3

Volume of any Prism V = BH

Volume of a Cylinder V = BHV = r 2 H

V = 4 cm6 cm()( 2)( )=301.6 cm346

Volume of Pyramids and Cones Experiment

V = 213.3 m36410

V = 1005.3in3815

Volume of a sphereV =4 r3

5 mr = V =
5523.6 m34 ( )35

Since a hemisphere is of a sphere its volume is the volume of the sphere.

Find the volume.V =( )( )( )32 cm3228

Find the volume.V = ( )( )( ) =18 in3433

V =()( 2)( ) =2309.07 in3Find the volume.715

V =( )( ) 3 = 64 in3Find the volume.643

V =( 2)( ) 369.74 in3Find the volume.8 2 = x2 + 32 x =x3 7.4 7.4

V =314.16 cm3Find the volume.( 2)( ) 35 12

V =4()( 3) 3904.78 cm3Find the volume.6

V =( )( ) 3366V = ( )( )( )666288 ft3+

V =97.9 m3( 2)( ) 32.55 5.1 V = ( )( )( )5.15.15.1

V =173.4 in3V =()( 2)( ) =2.57.5( 2)( ) 32.5 4 +

Find the volume of the sand.V =( 2)( ) 33.9 3.9 62.12 in3

1.Figure ABCDE has interior angle measures of 110, 90, 125, 130, and x. Find x.2.A regular polygon has 13 sides. Find the sum of the measures of the interior angles.3.Find the measures of one interior angle of a regular 22gon.4.What is the sum of the measures of one set of exterior angles of a 25gon?851980163.64360

5.What is the measure of each exterior angle of a regular octagon?
6.The measure of each exterior angle of a regular polygon is 36. How many sides does the polygon have?
7.Find the number of sides in a regular polygon if its interior angles are each 162.4510 sides20 sides

105a30s=a=n=106A = san

6.A polygon has interior angle measures of 120, 80, 135, 120, 100, and x. Find x.7.A regular polygon has 15 sides. Find the sum of the measures of the interior angles.8.Find the measure of one interior angle of a regular 24gon.9.What is the sum of the measures of one set of exterior angles of a 50gon?1652340165360

10.What is the measure of each exterior angle of a regular 40gon?
11.The measure of each exterior angle of a regular polygon is 7.2. How many sides does the polygon have?
12.Find the number of sides in a regular polygon if its interior angles are each 174.950 sides60 sides

126a30s=a=n=126A = san= 374

126a36s=a=n=125A = san8.3= 249

12xa22.5s=a=n=9.28A = san11.1= 408.512