Math 8 Unit 8Polygons and Measurement

Strand 4: Concept 4 Measurement

Strand 4: Concept 1 Geometric Properties PO 2. Draw three-dimensional figures by applying properties of eachPO 3. Recognize the three-dimensional figure represented by a netPO 4. Represent the surface area of rectangular prisms and cylinders as the area of their net.PO 5. Draw regular polygons with appropriate labels

PO 1. Solve problems for the area of a trapezoid.PO 2. Solve problems involving the volume of rectangular prisms and cylinders.PO 3. Calculate the surface area of rectangular prisms or cylinders. PO 4. Identify rectangular prisms and cylinders having the same volume.

Key TermsDef: Polygon: a closed plane figure formed by 3 or more segments that do not cross each other.

Def: Regular Polygon: a polygon with all sides and angles that are equal.

Def: Interior angle: an angle inside a polygon

Def: Exterior angle: an angle outside a polygon

Triangle3 Sides3 AnglesSum of Interior Angles 180Each angle measures 60 if regular.

Quadrilateral4 Sides4 AnglesSum of Interior Angles 360Each angle measures 90 if regular.

Pentagon5 Sides5 AnglesSum of Interior Angles 540Each angle measures 108 if regular.

Hexagon6 Sides6 AnglesSum of Interior Angles 720Each angle measures 120 if regular.

Heptagon7 Sides7 AnglesSum of Interior Angles 900Each angle measures 128.6 if regular.

Octagon8 Sides8 AnglesSum of Interior Angles 1080Each angle measures 135 if regular.

Nonagon9 Sides9 AnglesSum of Interior Angles 1260Each angle measures 140 if regular.

Decagon10 Sides10 AnglesSum of Interior Angles 1440Each angle measures 144 if regular.

Formula:

Example: Find the sum of the interior angles in the given polygon.

a. 14-gon •b. 20-gon 180(n-2)

Total = 2160 180●12

180(14-2)180(n-2)

Total = 3240 180●18180(20-2)

Example: Find the measure of each angle in the given regular polygon.

a. 16-gon

180(n-2)

Total = 2520 180●14

180(16-2)180(n-2)

Total = 1800 180●10

180(12-2)

b. 12-gon

2520 ÷16157.5

1800 ÷12150

Example: Find the length of each side for the given regular polygon and the perimeter.

a.) rectangle, perimeter 24 cm

b.) pentagon, 55 m

24 ÷ 46 cm

55 ÷ 511 m

Formula:

Example: Find the length of each side for the given regular polygon and the perimeter.

d. heptagon, 56 mm

c.) nonagon, 8.1 ft

8.1 ÷ 90.9 ft

56 ÷ 78 mm

Perimeter

Any shape’s “perimeter” is the outside of the shape…like a fence around a yard.

Evil mathematicians have created formulas to save you time. But, they always change the letters of the formulas to scare you!

Perimeter

Triangles have 3 sides…add up each sides length.

88

8

8+8+8=24The Perimeter is 24

To calculate the perimeter of any shape, just add up “each” line segment of the “fence”.

PerimeterA square has 4 sides of a fence

12 12

12

12

12+12+12+12=48

Regular PolygonsJust add up EACH segment

10

8 sides, each side 10 so 10+10+10+10+10+10+10+10=80

AreaArea is the ENTIRE INSIDE of a shapeIt is always measured in “squares” (sq. inch, sq ft)

Different Names/Same ideaLength x Width = Area

Side x Side = Area

Base x Height = Area

Notes 3-D ShapesBase: Top and/or bottom of a figure. Bases

can be parallel.Edge: The segments where the faces meet.Face: The sides of a three-dimensional shape.Nets: Are used to show what a 3-D shape would look like if we unfolded it.

PrismsHave RectanglesRectangles for facesNamed after the shape of their Bases

More Nets

by D. Fisher

Vertices (points)

Edges (lines)

Faces (planes)

6 6

99

55The base has 33 sides.

Vertices (points)

Edges (lines)

Faces (planes)

8

1212

66

The base has sides.44

Vertices (points)

Edges (lines)

Faces (planes)

1010

1515

77

The base has sides.55

Vertices (points)

Edges (lines)

Faces (planes)

1212

1818

88

The base has sides.66

Vertices (points)

Edges (lines)

Faces (planes)

1616

2424

1010

The base has sides.88

PyramidsHave TrianglesTriangles for facesNamed after the shape of their bases.

By D. Fisher

Vertices (points)

Edges (lines)

Faces (planes)

4 4

66

44The base has 33 sides.

Vertices (points)

Edges (lines)

Faces (planes)

55

88

55

The base has sides.44

Vertices (points)

Edges (lines)

Faces (planes)

66

1010

66

The base has sides.55

Vertices (points)

Edges (lines)

Faces (planes)

77

1212

77

The base has sides.66

Vertices (points)

Edges (lines)

Faces (planes)

99

1616

99

The base has sides.88

Name Picture Base Vertices Edges Faces

Triangular Triangular PyramidPyramid

Square Square PyramidPyramid

Pentagonal Pentagonal PyramidPyramid

Hexagonal Hexagonal PyramidPyramid

Heptagonal Heptagonal PyramidPyramid

Octagonal Octagonal PyramidPyramid

33 44 66 44

44 55 88 55

55 66 1010 66

66 77 1212 77

77 88 1414 88

88 99 1616 99

Any PyramidAny Pyramid nn n + 1n + 1 2n2n n + 1n + 1

Draw itDraw it

No No picturepicture

Name Picture Base Vertices Edges Faces

Triangular Triangular PrismPrism

Rectangular Rectangular PrismPrism

Pentagonal Pentagonal PrismPrism

Hexagonal Hexagonal PrismPrism

Heptagonal Heptagonal PrismPrism

Octagonal Octagonal PrismPrism

33 66 99 55

44 88 1212 66

55 1010 1515 77

66 1212 1818 88

77 1414 2121 99

88 1616 2424 1010

Any PrismAny Prism nn 2n2n 3n3n n + 2n + 2

Draw itDraw it

No No picturepicture

CylinderCirclesCircles for basesRectangle for side

Points of View

View point is lookingdown on the top ofthe object.

View point is lookingup on the bottom ofthe object.

View point is lookingfrom the right (or left)of the object.

Front View

Top View

Side View

Example 1 :

Bottom ViewBottom

FrontSide

SideFront

Top

Example 2 : Top

H

D

Front View

Left View

Example 3 : Top view

Left View Front View

Front View

Example 4

Left View

Top View

Bottom View

Surface AreaSurface Area: the total area of a three-dimensional figures outer surfaces. Surface Area is measured in square units (ex: cm2)

Rectangular Prism

SA=2lw +2lh + 2wh

l l

h

hh w

w

h w

wl

1. Find the surface area.

SA=2lw +2lh + 2wh

WL

H

SA=248 + 242 + 282SA= 64 + 16 + 32SA= 112 cm2

2. Find the surface area of a box with a length of 6 in, a width of 6 inches and a height of 10 inches.

SA=2lw +2lh + 2whSA=266 + 2610 + 2610

SA= 72 + 120 + 120SA= 312 cm2

Cylinder

r

h

r

h

r

SA = r2 + r2 + hC

C=2r

A=hC

So A=2rh

SA =2r2 + 2rh

Examples: 1. Find the surface area.

SA =2r2 + 2rh

SA = 2(5)2 + 2(5)(20)SA = 225 + 2100SA = 50 + 200 SA = 250 = 785 cm2

2. Find the surface area of a cylinder with a height of 5 in and a diameter of 18 in.

SA =2r2 + 2rh

SA = 2(9)2 + 2(5)(18)SA = 281 + 290SA = 162 + 180 SA = 342 = 1060.2 in2

VolumeVolume: The amount of space inside a 3D shape. Volume is measure in cubic units (ex: cm3)

Rectangular Prism

V=LWH

V = 842V = 64 cm3

CylinderV=r2h

V = 2(5)2(20)

SA = 1000 = 3140 cm3

V = 22520V = 2500

Triangular PrismV= ½ LWH V = ½ 244910

V = 5, 880 cm3

Surface Area or VolumeCovering a Triangular speaker box with carpet? Surface AreaFilling a triangular speaker box with foam? VolumeFilling a triangular box with M n M’s? Volume

Surface Area or VolumePainting the outside of a triangular prism? Surface AreaCovering a triangular piece of chocolate with paper? Surface AreaFilling a triangular mold with concrete? Volume