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Big Idea 2: Develop an understanding of and use formulas to determine surface areas and volumes of...
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Transcript of Big Idea 2: Develop an understanding of and use formulas to determine surface areas and volumes of...
Big Idea 2:
• Develop an understanding of and use formulas to determine surface areas and volumes of three-dimensional shapes.
Benchmarks
• MA.7.G.2.1: Justify and apply formulas for surface area and volume of pyramids, prisms, cylinders, and cones.
• MA.7.G.2.2: Use formulas to find surface areas and volume of three-dimensional composite shapes.
Vocabulary
• The vocabulary can easily be generated from the reference sheet and the Key.
• This will help you not only to review key vocabulary but the symbols for each word.
Vocabulary
• Take out the vocabulary sheet provided for you and fill in the second column with the definition for each word.
– Vocabulary Activity Sheet
• Next label the part image in the third column with the letter representing the corresponding vocabulary word. If there is no image draw one.
Review Perimeter
• Use the worksheets to review circumference and Pi
– Rolling a circle
– Archemedes estimation of Pi
• Use the following PowerPoint to review Perimeter
– Perimeter PowerPoint
Review Topics
GeoGebra activities for Area of Polygons and Circles
• Rectangles:– Area of a Rectangle
• Parallelograms:– Area of a Parallelogram
• Triangles:– Area of a Triangle
Review Topics
GeoGebra activities for Area of Polygons and Circles
• Trapezoids:– Area of a Trapezoid
• Circles:– Area of a Circles
Review Composite Shapes
• PowerPoint for discussing area and perimeter of composite figures.
– Composite Shapes PowerPoint
Side 2
Bottom
Back
Top
Side 1Front
Side 2
Bottom
Back
Top
Side 1Front
Length (L)Breadth (B)
Height (H)
Rectangular Solid
GeoGebra for a Cube
Bases
• Do the words Bottom and Base mean the same thing?
Base of a 3D Figure• Prism: a prism has 2 Bases and the bases, in all but a rectangular
prism, are the pair of non-rectangular sides. These sides are congruent, Parallel.
Bases
Triangular Prism
Base of a 3D Figure
Bases
CylinderGeoGebra Net for Cylinder
Base of a 3D Figure
Base
• Pyramid: There is 1 Base and the Base is the surface that is not a triangle.
Base of a 3D Figure• Pyramid: In the case of a triangular pyramid all sides are triangles.
So the base is typically the side it is resting on, but any surface could be considered the base.
Base
Net Activity
• Directions sheet
• Net Sheets
• Scissors
• Tape/glue
Building Polyhedra
GeoGebra Nets
• Net of a Cube
• Net of a Square Pyramid
• Net of a Cylinder
• Net of a Cone
• Net of an Octahedron
The net
w
w
w
w
b
h
hh
h
w
b
b b bb
h
h
h
h
?
?
?
Total surface Area =
Total surface Area
w
w
h
b
b b
h
h
b x h b x h
w x h
w x h
w x b
w x b
+ + + + +
= 2(b x h) + 2(w x h) + 2(w x b)
= 2(b x h + w x h + w x b)
Total surface Area
Nets of a Cube
• GeoGebra Net of a Cube
Activity: Nets of a Cube
• Given graph paper draw all possible nets for a cube.
• Cube Activity Webpage
Nets of a Cube
• Lateral Area is the surface area excluding the base(s).
Lateral Area
Net of a Cube
Lateral Area
Bases
Lateral Sides
Lateral Area
BasesLateral Surface
Net of a cylinder
Stations Activity
• At each station is the image of a 3D object. Find the following information:
– Fill in the boxes with the appropriate labels – Write a formula for your surface area– Write a formula for the area of the base(s)– Write a formula for the lateral area
Net handouts and visuals
• Printable nets– http://www.senteacher.org/wk/3dshape.php– http://www.korthalsaltes.com/index.html– http://www.aspexsoftware.com/
model_maker_nets_of_shapes.htm– http://www.mathsisfun.com/platonic_solids.html
• GeoGebra Nets– http://www.geogebra.org/en/wiki/index.php/
User:Knote
Volume
• The amount of space occupied by any 3-dimensional object.
• The number of cubic units needed to fill the space occupied by a solid
Volume Activity
• Grid paper
• Scissors
• 1 set of cubes
• Tape
Solid 1
Solid 2
Solid 3
Solid 3
Solids 4 & 5
• Circular Base
• Pentagon Base
Volume
• The number of cubic units needed to fill the space occupied by a solid.
1cm1cm
1cm
Volume = Base area x height
= 1cm2 x 1cm
= 1cm3
Rectangular Prism
• Volume = Base area x height= (b x w) x h = B x h
LL
L
• Total surface area = 2(b x w + w x h + bxh)
Comparing Volume
h
b
w
When comparing the volume of a Prism and a Pyramid we focus on the ones with the same height and congruent bases.
b
w
h
Comparing Volume
h
bw
w
b
w
b
h
Comparing Volume
h
l
Comparing Volume
h
l
Comparing Volume
b
w
h
h
b
w
Volume = B x h = b x w x h Volume = 1/3 (B x h) = 1/3 (b x w x h)
Prism Pyramid
2(LxB + BxH + LxH)
b x w x h
Rectangular Solid
6S2S3Cube
Sample net
Total surface area
VolumeFigureName
Volume formulas
• Prism and Cylinder– V=B x h
• Pyramid and Cone– V=1/3 (B x h)
Composite figure
8
1212
12