CUBE IN A CUBE
SURFACE AREA,
VOLUME &
CONSTRUCTION
Julie Pisarski 9B
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Julie Pisarski
Mr. Acre
Algebra, Geometry and transformations
10 January 2014
Cube in a Cube
Introduction
The freshman classes of 2017 are all individually preparing their own unique paper on an
origami project incorporating geometry skills they have acquired throughout the year. Through
the use of origami figures they must show the skills are ways to calculate surface area and
volume. This paper will show the construction process, step by step of the cube in a cube
origami project. It will show you the surface area of the inner and outer cube. It also contains the
volume of the inner cube and the volume of the space inside the outer cube.
Construction of the cube inside of a cube
Figure 1
Materials needed to make the entire cube in a cube; this includes 18 sheets of 6x6 inch paper.
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Construction of the Inner/Small Cube
1. Start with 1 piece of your 6x6 origami papers.
Figure 2
1 sheet of 6x6 paper
2. Fold the sheet in half with the color facing out
Figure 3
Sheet folded in half, this sheet is now a 3x6 simply because it is in half
3. Open paper back up, white facing up then fold each horizontal side to the middle crease
made in step 2
Figure 4
Opened sheet and folds to the center crease, color facing out
4. Flip paper over so the folds just made are facing the back.
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Figure 5
Sheet turned over, folds facing down, still 3x6 sheet
5. Take the right edge of the rectangle that has been formed and bring it up through the
upper right vertex.
Figure 6
The right edge up to the top edge. By doing this you create more measurements. The first top
section (red) is now 3 inches because we are looking at only the top. The second new
measurement (purple) is also 3 inches because it is a square that’s formed which means all the
sides will be the same. The last new measurement (black) is 3√2. This comes from the fact that
when dealing with a 45 45 90 triangle the hypotenuse is a√2.
6. Rotate 180 degrees and repeat step 5.
Figure 7
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Both right edges to top edge. The bottom (green) measurement is 3 inches still because of the
square (figure6) same method used with the purple, also 3 inches. Again using the same method
of looking at the square 3 inches is found for the orange measurement. Black is 3√2 (figure 6)
7. Take the top edge of the newly formed parallelogram to the vertical line segment formed
by the folds.
Figure 8
Top edge to vertical line segment
8. Rotate 180 degrees and repeat
Figure 9
Completed face of inner cube. The crease in the middle (purple) still remains 3 inches because it
hasn’t been folded. However a new measurement (blue) is formed now that the black side in
figure 7 has been folded in half. This creates the new measurement of 1.5√2
9. Repeat steps 1-8 until you have 6 faces
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Figure 10
All 6 completed faces
10. Each triangle coming out of the square is called a connector, the square with the 2
pockets are called flaps. Each flap will contain 2 connectors, each from a different face.
Starting with 2 faces slide the connector into the flap.
Figure 11
Connector into flap
11. Repeat step 10 until your cube is finished
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Figure 12
Finished inner cube
Construction of large/outer cube
1. Start with 1 6x6 inch square
Figure 13
Square sheet of paper, 6x6, color up
2. Fold the square in half color out
Figure 14
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Fold in half to create a crease down the center. Unfold
3. Open up the paper, color up and fold the edges to the fold line
Figure 15
Flipped over and folded to the center. The paper is now a 3x6
4. Fold upper right corner; Rotate 180 degrees and repeat
Figure 16
Upper right corner and lower left folded up. To find the measurements you can look at the new
folds and see that (black) is half of the side, the side that has already been folded in half so now it
is essentially ¼ of the measurement which comes out to be 1.5 inches. Using the triangle method
you can see that it’s a 45 45 90 triangle looking at it. Now it’s shown the measurement (blue) is
1.5√2
5. Flip the paper over so the open edges are against the table, and the fold line is visible
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Figure 17
Paper flipped, opening towards the bottom.
6. Take the right edge to the top edge through the upper right vertex. Rotate 180 degrees
repeat
Figure 16
Right edge to the top edge through the upper right vertex. To find the measurements of black
refer to figure 16. To find the red side you can look back to the beginning when the paper was
folded in half and this side is now the same length as the opposite side and it also forms a square
making the red side 3 inches. For the blue side looking at the 45 45 90 triangle method you can
take the side length 3 inches (red) and make that into 3√2 for the length.
7. Fold sides together color out
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Figure 17
Folded together color out. The bottom measurement (blue) is the same 3√2 that hasn’t
changed at all. Looking up at the top measurement (green) it is half of the bottom so it
becomes 1.5√2
8. Fold edges as shown, flip and repeat.
Figure18
Edges folded toward center. The red side is now 3 inches. The blue remains the same 3√2
inches because it hasn’t changed at all and the black is 1.5 inches.
9. Repeat steps 1-8 until you have 12 pieces
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Figure 19
All sides completed
10. To start assembly begin with 2 pieces
Figure 20
2 starting pieces
11. Insert the yellow triangle into the pink side triangle
Figure 21
Assembly of the corner
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12. Continue assembling triangles accordingly
Figure 22
Finished cube
*Note* the cubes coordinate in a way that the openings of the outer cube (the square
holes) are the same size of the faces of the inner cube.
Figure 23
Shows the congruency of the face of the inner cube with the opening of the outer cube
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Figure 24
Finished cube in a cube
Surface Area
By finding the surface area of the large cube and adding it to the surface area of the small
cube you can find the total surface area of the 2 cubes combined. Keeping in mind the surface
area of the large cube does not include the hole in each of the faces. One approach to finding the
surface area is to find the area of one of the triangles. Since we know that the original sheet we
started with was a 6x6 after folding is would be a 3x6. Then after folding the corners up it
becomes 1.5 inches (refer to figure 25 and 26 for reference)
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3√2 inches
Figure 27
Measurements of the sides of one piece of the outer cube before assembly (see
instructions for measurement recaps
To find the legs you must now divide by √2. When you do that it will come out to be .75√2. Now
that you have the leg you can plug them into the area formula of a triangle which is
A=1/2(base*height). Which will come out looking like A=1/2*(3/4√2)*(3/4√2). If you simplify
it down it becomes A=1/2*(9/16√4). The ½ and the √4 will cancel each other out. Then this will
leave you with A=9/16inches squared or .5625 inches². When multiplied by 8 you can get the
surface area of one of the columns and when multiplied by 6 you can get the surface area of one
of the corners.
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Corner
Column
In addition, each column has 8 triangles; these are the triangles we just found the area of.
Since the area of one of the triangles is .5625 you can multiply that by 8 since there are 8 of the
same triangles in each column. When done so (.5625)*8 you will get the surface area of 1
column to be 4.5 inches². When looking at the cube notice there are 12 columns so you can then
multiply 4.5 by 12 to get the area of the columns. This calculates to 54 inches²
When looking at the cube there are also 8 corners, each containing 6 of those same
triangles with an area of .5625 Using the same process done in the columns, next do the same but
instead of multiplying by 8 multiply by 6 (since the 6 equilateral triangles all have the .5625
area) in each corner. When doing this process the surface area of the corner ends up coming out
to 3.375 inches². Given the cube has 8 corners you now must take our surface area of that 1
corner and multiply it by 8 to get the surface area of all the corners. This ends up being 27; the
area of the corners is 27 inches²
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Now that you have the area of all 12 columns and all 8 corners add them up to get the
total surface area of the outer cube; 81 inches²
Inner cube
Using previous information that the original sheet of paper was 6x6 when folded in half it
becomes 3x6. The paper will then get folded another time (step 7 in inner cube) and then the side
length is 1.5. When looking from a triangle method it is clear that the side length is 1.5√2
Figure 28
Side length triangle
Now knowing the one side of a triangle the area formula of a triangle can now be used.
A=1/2*(base*height). When plugged in looking like A=1/2(1.5√2)*(1.5√2). Simplifying it to be
A=1/2(2.25√4). The ½ and the √4 will cancel each other out which makes the area of the
triangle to be 2.25 inches² Since there are 2 triangles in each square (face) each face ends up
being 4.5 inches² The cube has 6 faces so the next step is to multiply the 4.5 by 6. This ends up
being 27. The surface area of the inner cube is 27 inches²
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To find the total surface area, the large cube must be added to the small cubes surface
area. 81+27+108 inches²
The total surface areas of the 2 cubes are 108 inches²
Volume:
Volume formula: V= length * width * height
Space between the small and large cube: subtract the volume of the small cube from the
volume of the large cube
Start by finding the volume of the smaller cube by using the volume formula. Since it’s a
cube the length the width and the height will all be the same. This is 1.5√2. 1.5√2 * 1.5√2 *
1.5√2 = 3.375*2√2. Multiplying 3.375 by 2 will get you 6.75√2inches³ which is the volume of
the inner cube.
The outer cube uses the same method. V=l*w*h. the length width and height if the cube
are all 3√2 again because a cube has the same measurements. 3√2 * 3√2 * 3√2 = 27 * 2√2.
Multiply 27 by 2 and get 54√2inches³. It is not finished yet because you must subtract the
volume of the small cube from the large cube because the inner cube takes up space. When done
54√2 – 6.75√2 = 47.25√2 inches³ which is the completed volume of the outer cube.
Conclusion:
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In conclusion the surface area of both the cubes is 108 inches ² coming from the smaller
cube being 27 inches² and the larger cube being 81 inches². The volume of the outer cube is
47.2√25inches³ when the inner cube is inside. Without the inner cube inside the volume is
54√2. What is noticeable with this data is that the inner cube is 8 times smaller than the outer
cube and that’s the connection they share. The volume of the inner cube is 6.75√2inches³
exactly 8 times less than the outer.