math 8 unit 8 polygons and measurement strand 4: concept 4 measurement strand 4: concept 1 geometric...
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Triangle 3 Sides 3 Angles Sum of Interior Angles 180 Each angle measures 60 if regular.TRANSCRIPT
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