3-D Geometry - Center For Teaching &...
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Slide 1 / 138 3-D Geometry
Transcript of 3-D Geometry - Center For Teaching &...
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3-D Geometry
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Table of Contents
Surface Area· Prisms· Pyramids· Cylinders
· Prisms and CylindersVolume
· Pyramids, Cones & Spheres
Nets
Click on the topic to go to that section
More Practice/ Review· Spheres
3-Dimensional Solids
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The following link will take you to a site with interactive 3-D figures and nets.
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Polyhedron A 3-D figure whose faces are all polygons
Polyhedron Not Polyhedron
Sort the figures into the appropriate side.
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3-Dimensional SolidsCategories & Characteristics of 3-D Solids:
Prisms1. Have 2 congruent, polygon bases which are parallel to one another2. Sides are rectangular (parallelograms)3. Named by the shape of their base
Pyramids1. Have 1 polygon base with a vertex opposite it2. Sides are triangular3. Named by the shape of their base
click to reveal
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3-Dimensional SolidsCategories & Characteristics of 3-D Solids:
Cylinders1. Have 2 congruent, circular bases which are parallel to one another2. Sides are curved
Cones1. Have 1 circular bases with a vertex opposite it2. Sides are curvedclick to reveal
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3-Dimensional Solids
Vocabulary Words for 3-D Solids:
Polyhedron A 3-D figure whose faces are all polygons (Prisms & Pyramids)
Face Flat surface of a Polyhedron
Edge Line segment formed where 2 faces meet
Vertex (Vertices) Point where 3 or more faces/edges meet
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Sort the figures.If you are incorrect, the figure will be sent back.
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For each figure, find the number of faces, vertices and edges.
Can you figure out a relationship between the number of faces, vertices and edges of 3-Dimensional Figures?
Name Faces Vertices Edges
Cube 6 8 12
Rectangular Prism
6 8 12
Triangular Prism
5 6 9
Triangular Pyramid
4 4 6
Square Pyramid
5 5 8
Pentagonal Pyramid
6 6 10
Octagonal Prism
10 16 24
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Euler's Formula
F + V - 2 = E
The number of edges is 2 less than the sum of the faces and vertices.
click to reveal
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NetsNets are two-dimensional drawings that represent the surface area of three-dimensional shapes.
There is more than one way to draw a net for a cube, however not all nets can be folded into a cube...
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Cut arrangements of 6 squares out of grid paper.
How many different arrangements can be folded into a cube?
Hint:You just saw one arrangement that works and one that does not.
see next page for answers...
Nets - Flat Patterns Activity Part 1
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click to reveal the cube flat patterns
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Cut an arrangement out of grid paper to create a rectangular prism that is not a cube .
Now make another flat pattern for the same rectangular prism.
Try each pattern out by folding it into a box.
What are the dimensions of each face of your rectangular prism?
Nets - Flat Patterns Activity Part 2
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Nets for prisms will have rectangular faces and two bases for which the shape is named.
Notice the two triangles areopposite from one another (bases).
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Volume
Volume
- The amount of space occupied by a 3-D Figure - The number of cubic units needed to FILL a 3-D Figure (layering)
Label
Units3 or cubic units
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VolumeVolume of Prisms & Cylinders:
Area of Base x Height
Area Formulas:
Rectangle = lw or bh
Triangle = bh or 2 Circle = r2
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(bh)
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Find the Volume.
5 m
8 m
2 m
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Find the Volume.
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10 yd
9 yd
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14 Find the Volume.
7 in 1 5
1 in 1 2
4 in
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15 Find the volume of a rectangular prism with length 2 cm, width 3.3 cm and height 5.1 cm.
Ans
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17 Find the volume.
21 ft
42 ft
50 ft47 ft A
nsw
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18 A box-shaped refrigerator measures 12 by 10 by 7 on the outside. All six sides of the refrigerator are 1 unit thick. What is the inside volume of the refrigerator in cubic units?
HINT: You may want to draw a picture!
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19 Find the volume.
6 m
10 m
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20 Which circular glass holds more water?
A Glass A having a 7.5 cm diameter and standing 12 cm high
B Glass B having a 4 cm radius and a height of 11.5 cm
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21 What is the volume of the largest cylinder that can be placed into a cube that measures 10 feet on an edge?
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22 A circular garden has a diameter of 20 feet andis surrounded by a concrete border that has a width ofthree feet and a depth of 6 inches. What is the volume of concrete in the path? Use 3.14 for .#
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Demonstration comparing volume of Cones & Spheres with volume of
Cylinders
click to go to web site
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Volume of a Cone
(Area of Base x Height) 3
(Area of Base x Height) 1 3click to reveal
A cone is 1/3 the volume of a cylinder with the same base area (B) and height (h).
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V = 2/3 (Volume of Cylinder)
r2 h( )2/3 V=or
V = 4/3 r3
Volume of a Sphere
click to reveal
A sphere is 2/3 the volume of a cylinder with the same base area (B) and height (h).
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How much ice cream can a Friendly’s Waffle cone hold if it has a diameter of 6 in and its height is 10 in?
(Just Ice Cream within Cone. Not on Top)Volume and Mass used in portion control. $$$
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23 Find the volume.
4 in
9 in
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24 Find the Volume
5 cm8 cm
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25 What is the volume of a sphere with a radius of 8 ft?
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26 What is the volume of a sphere with a diameter of 4.25 in?
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Volume of a Pyramid
(Area of Base x Height) 3
(Area of Base x Height) 1 3click to reveal
A pyramid is 1/3 the volume of a prism with the same base area (B) and height (h).
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Pyramids are named by the shape of their base..
The volume is a pyramid is 1/3 the volume of a prism with the same base area(B) and height (h).
V = Bh13
=5 m
side length = 4 m
V = Bh13
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27 Find the Volume of a triangular pyramid with base edges of 8 in, base height of 4 in and a pyramid height of 10 in.
8 in
10 in
4 in
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28 Find the volume.
8 cm
7 cm
15.3 cm
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Surface Area
The sum of the areas of all outside surfaces of a 3-D figure.
To find surface area, you must find the area of each surface of the figure then add them together.
6 in
3 in8 in
What type of figure is pictured?
How many surfaces are there?
How do you find the area of each surface?
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Surface Area
6 in
3 in8 in
Bottom Top Left Right Front Back SUM 3 6 8 24 x 8 x 3 x 6 24 24 24 18 18 48 48 18 18 48 +48 180 in2
3 in8 in
6 in
8 in
6 in
3 in
6 in
3 in8 in
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Arrangement of Unit Cubes
Find all the possible ways that 24 unit cubes can be arranged into a rectangular prism. Give each arrangements dimensions and surface area. Also make a sketch.
Length Width Height Volume Surface Area
Sketch
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Arrangement of Unit Cubes
Find all the possible ways that 64 unit cubes can be arranged into a rectangular prism. Give each arrangements dimensions and surface area. Also make a sketch.
Length Width Height Volume Surface Area
Sketch
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29 Which arrangement of 27 cubes has the least surface area?
A 1 x 1 x 27
B 3 x 3 x 3
C 9 x 3 x 1
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30 Which arrangement of 12 cubes has the least surface area?
A 2 x 2 x 3
B 4 x 3 x 1
C 2 x 6 x 1
D 1 x 1 x 12
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31 Which arrangement of 25 cubes has the greatest surface area?
A 1 x 1 x 25
B 1 x 5 x 5
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32 Which arrangement of 48 cubes has the least surface area?
A 4 x 12 x 1
B 2 x 2 x 12
C 1 x 1 x 48
D 3 x 8 x 2
E 1 x 2 x 24
F 4 x 3 x 4
G 1 x 8 x 6
H 4 x 2 x 6
I 3 x 16 x 1
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Find the surface area of a rectangular shoe box that has a length of 12 inches, a width of 6 inches and a height of 5 inches.
Top/Bottom Front/Back Left/Right Surface Area 12 12 6 144 x 6 x 5 x 5 120 72 60 30 + 60 x 2 x 2 x 2 324 in2 144 120 60
5 in
6 in
12 in
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Find the Surface Area.1. Draw and label ALL faces2. Find the correct dimensions for each face3. Calculate the AREA of EACH face4. Find the SUM of ALL faces5. Label Answer
5 yd
6 yd
4 yd
9 yd
5 yd
go on to see steps
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Triangles Bottom Rectangle 4 9 x 6 x 6 24 / 2 = 12 54 x 2 24 Total
SurfaceArea 24 54+ 90 168 yd2
5 yd
6 yd
4 yd9 yd
5 yd
9 yd
5 yd
5 yd
6 yd
4 yd
Left/Right Rectangles(Same size since isosceles) 5 x 9 45 x 2 90
9 yd
5 yd
5 yd
6 yd
4 ydClick on box to
reveal steps.
Click on box to reveal steps.
Click on box to reveal steps.
Click on box to reveal steps.
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Find the Surface Area.1. Draw and label ALL faces2. Find the correct dimensions for each face3. Calculate the AREA of EACH face4. Find the SUM of ALL faces5. Label Answer
9 cm
6 cm 11 cm
9 cm
9 cm
go on to see steps
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Triangles Rectangles 9 (All 3 are the same size X6 since equilateral) 54 / 2 = 27 9 x 2 x 11 54 99 x 3 297
Surface Area 297+ 54 351 cm2
9 cm
6 cm 11 cm
9 cm
9 cm
9 cm
6 cm 11 cm
9 cm
9 cm
Click on box to reveal steps.
Click on box to reveal steps.
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39 Find the surface area of the shape below.
21 ft
42 ft
50 ft
47 ft
1. Draw and label ALL faces2. Find the correct dimensions for each face3. Calculate the AREA of EACH face4. Find the SUM of ALL faces5. Label Answer
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7 cm 3 cm
4 cm
15 cm
6 cm
5 cm
Find the Surface Area.
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Trapezoids 10 + 6 16 x 4 64 / 2 = 32 x 2 64
Bottom Rectangle 6 x 15 90
Top Rectangle 10 x 15 150
Side Rectangles 5 x 15 75 x 2 150
Surface Area 64 90 150 + 150 454 cm2
7 cm 3 cm
4 cm
15 cm
6 cm
5 cm
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40 Find the surface area of the shape below.
8 cm 6 cm
10 cm
9 cm
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41 Find the surface area of the shape below.
10 cm2 cm
6 cm
10 cm
6 cm
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Surface Area of Pyramids
What is a pyramid?
Polyhedron with one base and triangular faces that meet at a vertex
How do you find Surface Area?
Sum of the areas of all the surfaces of a 3-D Figureclick to reveal
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8 cm
8 cm
17.5 cm
Find the Surface Area.
go on to see steps
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Find the Surface Area.
Bottom Rectangle 8 x 7 56
Front/Back Triangles
Left/RightTriangles
Surface Area 56 140+ 140 336 cm2
8 cm
8 cm
17.5 cm
8 cm
17.5 cm
8 cm
8 cm
17.5 cm
8 cm
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Find the surface area of a square pyramid with base edge of 4 inches and triangle height of 3 inches.
4 in
3 inBase 4x 4 16
Triangles Surface Area 16+ 24 40 in2
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Find the surface area.
Be sure to look at the base to see if it is an equilateral or isosceles triangle (making all or two of the side triangles equivalent!).
Base Triangles(all equal)
Surface Area 7+ 36 43 in2
4 in4 in
4 in
6 in
3.5 in
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42 Which has a greater Surface Area, a square pyramid with a base edge of 8 in and a height of 4 in or a cube with an edge of 5 in?
A Square Pyramid
B Cube
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43 Find the Surface Area of a triangular pyramid with base edges of 8 in, base height of 4 in and a slant height of 10 in.
8 in8 in
8 in
10 in
6.9 in
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44 Find the Surface Area.
9 m9 m
12 m
11 m
6.7 m
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How would you find the surface area of a cylinder?
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Notice the length of the rectangle is actually the circumference of the circular base.
Steps1. Find the area of the 2 circular bases.2. Find the area of the curved surface (actually, a rectangle).3. Add the two areas.4. Label answer.
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Radius
Curved Side = Circumference of Circular Base
HEIGHT
HEIGHT
Radius
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Area of Circles = 2 ( )Area of Curved Surface = Circumference Height d Height
2 r2 + dh
2 r2 + 2 rh-or-
r2
Radius
Curved Side = Circumference of Circular Base
HEIGHT
HEIGHT
Radius
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Find the surface area of a cylinder whose height is 14 inches and whose base has a diameter of 16 inches.
14 in
16 in Area of Circles = 2 ( r2 ) = 2 ( 82) = 2 (64 ) = 128 = 401.92 in2
Area of Curved Surface = Circumference Height = d Height = (16)(14) = 224 = 703.36 in2
Surface Area = 401.92 + 703.36 = 1105.28 in2
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45 Find the surface area of a cylinder whose height is 8 inches and whose base has a diameter of 6 inches.
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46 Find the surface area of a cylinder whose height is 14 inches and whose base has a diameter of 16 inches.
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47 How much material is needed to make a cylindrical orange juice can that is 15 cm high and has a diameter of 10 cm?
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48 Find the surface area of a cylinder with a height of 14 inches and a base radius of 8 inches.
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49 A cylindrical feed tank on a farm needs to be painted. The tank has a diameter 7.5 feet and a height of 11 ft. One gallon of paint covers 325 square feet. Do you have enough paint? Explain.
Yes
No
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A sphere is the set of all points that are the same distance from the center point.
Like a circle, a sphere has a radius and a diameter.
You will see that like a circle, the formula for surface area of a sphere also includes pi.
Radius
Surface Area of a Sphereclick to reveal
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If the diameter of the Earth is 12,742 km, what is its surface area?
12,742 km
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Try This:
Find the surface area of a tennis ball whose diameter is 2.7 inches.
2.7 in
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50 Find the surface area of a softball with a diameter 3.8 inches.
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51 How much leather is needed to make a basketball with a radius of 4.7 inches?
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52 How much rubber is needed to make 6 racquetballs with a diameter of 5.7 inches?
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53 Find the volume.
15 mm
8 mm
22 mm
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54 Find the volume of a rectangular pyramid with a base length of 2.7 meters and a base width of 1.3 meters, and the height of the pyramid is 2.4 meters.
HINT: Drawing a diagram will help!
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55 Find the volume of a square pyramid with base edge of 4 inches and pyramid height of 3 inches.
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56 Find the Volume
9 m9 m
12 m
11 m
6 m
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57 Find the Volume
14 ft
21 ft
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58 Find the Volume
8 in
6.9 in
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59 Find the Volume
4 ft
7 ft
8 ft
9 ft
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60 A cone 20 cm in diameter and 14 cm high was used to fill a cubical planter, 25 cm per edge, with soil. How many cones-ful of soil were needed to fill the planter?
20 cm
14 cm
25 cm
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61 Find the surface area of the cylinder.
Radius = 6 cm and Height = 7 cm
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62 Find the Surface Area.
11 cm
11 cm
12 cm
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63 Find the Surface Area.
7 in8 in
9 in
9 in
2 in
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64 Find the Volume.
7 in8 in
9 in
9 in
2 in
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65 A rectangular storage box is 12 in wide, 15 in long and 9 in high. How many square inches of colored paper are needed to cover the surface area of the box?
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66 Find the surface area of a square pyramid with a base length of 4 inches and slant height of 5 inches.
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Name a 3-D Figure that is not a polyhedron.
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Define the following terms:
Surface Area
Volume
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Name a 3-D figure that has 6 rectangular faces.
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67 Find the volume.
40 m
70 m
80 m
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68 A teacher made 2 pair of foam dice to use in math games. Each cube measured 10 in on each side. How many square inches of fabric were needed to cover the 2 cubes?
