FILLING & WRAPPING 1.3
Learning Target: I will determine how to design a box with
the least packaging material, given the volume of a box.
HW: Finish the FW Investigation 1 Pg. 7 and
Correct with the Zaption video: FW 1.3
*WDYE Retake this Thursday, 3/24 in Math Lab
Warm-up:
What is the Surface Area of this Kleenex Box?
Dimensions: 21.5 cm x 10.5 cm x 5cm
Height = 5 cm
Width = 10.5 cm
Length = 21.5 cm
Warm Up
Kleenex Box -Dimensions: 21.5 cm x 10.5 cm x 5cm
Height = 5 cm
Width = 10.5 cm
Length = 21.5 cm
Surface Area:
Area of Front face = 21.5 x 5 = 107.5
Area of Top face = 21.5 x 10.5 = 225.75
Area of Right face = 10.5 x 5 = 52.5
Total Surface Area = 2 (107.5 + 225.75 + 52.5) = 771.5 sq. cm
What does it mean to be
3-dimensional?
• Lines are one-dimensional. They only have length.
• Polygons and circles are 2-dimensional because they
have 2 dimensions.
Width
Circumference
Length
Radius
Height
Width or Length
• Polyhedrons and spheres are 3-dimensional. They
have 3 dimensions.
Height
Length
Width
Nets
• A net is a two-dimensional shape that can be
folded into a three-dimensional shape.
Examples:
Our online tool can help you see how nets fold into rect. prisms.
Surface Area Recap
• Always label surface area as: units squared or
units2 because surface area only measures
two dimensions (length and width).
• Different ways to solve for surface area:
– Solve for the area of all six faces and add them all
together. OR
– Solve for the area of three of the faces (for
example, top, front, and right), add the areas
together, and multiply by two (because there are
two pairs of faces that are equivalent) .
Volume Recap
• Volume is always labeled as:
– units cubed or units3 because we use three
dimensions to solve for volume.
• To find the volume of a polyhedron:
– Count how many inch cubes fit in the box. OR
– Find the area of the base and multiply it by the
height of the polyhedron.
• Volume = Bh (B = area of base , h = height) OR
– Volume of a Rectangular Prism =
• Length x Width x Height
p. 5
You discovered yesterday that 24 blocks can be packaged in different
ways that use varying amounts of packaging material. By using less
material, a company can save money, reduce waste, and conserve
natural resources. Which rectangular arrangement of cubes uses the
least amount of packaging material?
Find All Possible Arrangements
• Group Task:
– Given 8 cubes find all possible arrangements for a
rectangular prism.
– Fill in the packet as you are working.
– Find the volume for each arrangement.
– Find the surface area for each arrangement.
– Think about the following:
• Which arrangement gives you the greatest SA?
• Which arrangement gives you the least SA?
Possible Arrangements of 8 cubes
Length
Width
Height
Volume
Surface Area
8 in
4 in
2 in
1 in
2 in
2 in
1 in
1 in
2 in
8 in³
8 in³
8 in³
34 in²
28 in²
24 in²
Does it matter which order the dimensions are in? No!
• Is (8 in x 1 in x 1 in) the same rectangular prism as
(1 in x 1 in x 8 in), or (1 in x 8 in x 1 in) ? Yes!
Possible Arrangements of 8 cubes
Length
Width
Height
Volume
Surface Area
8 in
4 in
2 in
1 in
2 in
2 in
1 in
1 in
2 in
8 in³
8 in³
8 in³
34 in²
28 in²
24 in²
1. Which arrangement had the most surface area and
used the most packaging material for 8 cubes?
8 in x 1 in x 1 in
2. Which arrangement had the least surface area and
used the least packaging material for 8 cubes?
2 in x 2 in x 2 in
a. What is the name of that prism? cube
Find All Possible Arrangements
• Group Task:
– Find all possible arrangements for 27 cubes .
– Fill in the packet as you are working.
– Find the volume for each arrangement.
– Find the surface area for each arrangement.
– Think about the following:
• Which arrangement gives you the greatest SA?
• Which arrangement gives you the least SA?
p.6
Possible Arrangements of 27 cubes
Length
Width
Height
Volume
Surface Area
27 in
9 in
3 in
1 in
3 in
3 in
1 in
1 in
3 in
27 in³
27 in³
27 in³
110 in²
78 in²
54 in²
Does it matter which order the dimensions are in? No!
27 x 1 x 1
1 x 27 x 1
1 x 1 x 27
Same SA and Volume!
3. Which arrangement (set of dimensions) used the least packaging
material for 27 cubes?
3 in x 3 in x 3 in
a. What is the name of this type of prism?
cube
4. Without solving for the surface area, how could you tell
simply by looking at the dimensions which prism will have
the least amount of surface area?
The shape closest to a cube has the least amount of surface area.
5. The dimensions for a box with 12 unit cubes are listed
below.
12 x 1 x 1
4x3x1
6x2x1
3x2x2
a. Which set of dimensions will require the least
amount of material or have the smallest surface area?
How do you know?
3x2x2
Because they are the closest dimensions to a cube.
FILLING & WRAPPING 1.3
Did I meet my Learning Target?
I will determine how to design a box with the least packaging
material, given the volume of a box.
HW: Finish the FW Investigation 1 Pg. 7 and
Correct with the Zaption video: FW 1.3
*Remember:
WDYE Retake this Thursday, 3/24 in Math Lab