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Cube in a Cube Project
Cubes are one of the most impressive shapes around. First of all they are one of the five
platonic shapes that have one regular polygon as faces, and have the same amount of faces
meeting at each vertex. Cubes have twelve sides, eight vertices, six faces. Cubes can be found in
everyday life as dice, as some types of crystals, in sugar cubes, and the Rubik’s Cube ®. Cubes
are prisms but prisms are not cubes just like how squares are rectangles but rectangles are not
squares. This shows that cubes are very interesting and that cubes are very important in geometry
because of there properties and apperances in everyday life. Cubes have volume and surface
area. How could you find the volume and surface area of cube models? The volume of cube
models can be found by finding a side length of a cube and then using it in common cube
formulas such as side^3 for volume and 6(base) for surface area.
Cubes can also be made with origami paper. The directions below show how to create a
larger cube that can fit around a smaller cube. A model of cubes can be very beneficial to people
learning geometry because it shows how to get volume and surface area in a hands-on
experience. To make them, six pieces of paper of three different colors are required.
1. Start off with a piece of origami paper color-side down. Fold the paper into fourths.
Figure 1. Step 1
Figure 1 shows how a piece of origami paper is folded into fourths color-side down
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2. Fold point B to point A, and fold point D to point C.
Figure 2. Step 2
Figure 2 indicates with lettered points where and how to fold by stating which point
should be pulled to another.
3. Fold point A to point B, and fold point C to point B.
Figure 3. Step 3
Figure 3 indicates with lettered points where and how to fold by stating which point
should be pulled to another.
4. This is the finished look. Reapeat steps 1-3 five more times to end up with six units.
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Figure 4. Step 4
Figure 4 shows how a complete unit should look like and also shows that six units are
needed of three different colors.
5. Take two units of the same color and one of a different color. Place the noses of the two,
alike units in the face of the other unit. After doing so the face of the other unit should
have four triangles.
Figure 5. Step 5
Figure 5 shows how to put the units together by placing a nose in a face and clearly
represents the text from step 5.
6. Take another unit of the same color as the unit with two noses in its face, so it ends up
two pairs of units with the same color. Take another nose of a unit and stick it in the face
of the new unit. Then curl around the last nose, and put it in the face of the new unit.
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Figure 6. Step 6
Figure 6 shows how to connect four units together in and clearly represents the text from
step 6.
7. Take the two last units and place them over the openings and place the unused noses of
the other units into them.
Figure 7. Step 7
Figure 7 shows how to add the last two units to the formation and gives a visual
interpretation of step 7.
8. Finally place the the rest of the unused noses into the unused faces.
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Figure 8. Step 8
Figure 8 shows the finished look of the smaller cube.
9. Start off with a piece of origami paper color-side up. Fold the paper into fourths.
Figure 9. Step 9
Figure 9 shows how a piece of origami paper is folded into fourths color-side up.
10. Fold the top right corner and the bottom left corner.
Figure 10. Step 10
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Figure 10 shows how the top right and bottom left corners should be folded.
11. Flip the paper over and fold point B to point A, fold point D to point C. Then fold paper
along line segment AB.
Figure 11. Step 11
Figure 11 shows points on the paper which are used to indicate certain folds. In addition
Figure 11 shows on what line to fold on an incomplete unit.
12. Fold both indicated squares in half diagonaly.
Figure 12. Step 12
Figure 12 shows how and where to fold which squares, verbalized in step 12, and
indicates the finished look of the unit.
13. Repeat steps 9-12 eleven more times to end up with twelve units with four of each color.
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Figure 13. Step 13
Figure 13 shows the twelve units needed of three different colors.
14. Take two units of different colors and place the nose of one into the face of another.
Figure 14. Step 14
Figure 14 shows visually how to connect two units together by placing the nose of one in
the face of another.
15. Then take a third unit of a different color and place an open nose in its face and put its
nose in an open face. Each end of every unit connects two other units by having a nose in
its face and putting its nose in another face.
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Figure 15. Step 15
Figure 15 shows with images how to connect a third unit to two connected units to create
a corner.
16. Repeat steps 14-15 three more times to end up with four corners.
Figure 16. Step 16
Figure 16 indicates that four corner need to be made to proceed to the next step.
17. Take two corners and attach them to each other so parrallel sides are the same color.
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Figure 17. Step 17
Figure 17 shows how to connect two corners together so parrallel colors are the same to
create a correct face.
18. Repeat step 17 once to end up with two faces that are alike in color.
Figure 18. Step 18.
Figure 18 visually shows that two faces should be formed and should look alike.
19. Connect the two faces together so that all faces opposite from each other are alike. Make
sure the each end of every unit is connected to two other units.
Figure 19. Step 19
Figure 19 shows how the two faces are connected and how the bigger cube looks when
finished.
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20. Finally place the smaller cube in the bigger cube.
Figure 20. Step 20
Figure 20 shows how to place the smaller cube in the bigger by pushing the smaller cube
through a gap in the bigger cube.
To get the volume and surface area of a three dimensional object it is best to first know
what volume and surface area are. Volume of a three dimensional shape is the space that the
shape takes up. Surface area is the area of all of the two dimensional faces of a three dimensional
shape added up. Volume will always be units ^3, and surface area will always be units^2. With
that in mind most volumes are found by base * height, where the base is a face and the height is a
perpendicular line segment to the base and is not on the same plane as the base. In addition most
surface areas area found by (perimeter * height) + 2(base). Neither of these formulas is true in all
cases, and an example are pyramids. The formula for the volume of a cube and surface area is
different, too. The formula for volume of a cube is side^3 which is true because all the sides of a
cube are the same, and it is really the same as base * height. The formula for the surface area of
cubes is 6(base) because all faces of any cube are alike, and there are six faces in a cube. The
volume and surface area formula are used in the figures below.
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Figure 21. Formulas for Surface Area and Volume of Small Cube
Figure 21 is a proof that shows how to the formulas for the volume and surface area were
derived. It starts off as a side of origami paper as x, shows how the height or segment FG was
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found, and then creates a new formula for volume and surface area of the small cube. The bottom
left picture, under the proof, shows where the height of an origami unit would be if it was folded
then unfolded. The bottom right picture shows the small cube, its base, its height, and where
segment FG could be at. The formula for surface area is ¾(x)^2 and the formula for volume is
1/32(x)^3√2.
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Figure 22. Formulas for Surface Area and Volume of Big Cube
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Figure 22 shows how the formulas for surface area and volume of the big cube were
derived. The formulas were derived from a side of origami paper being x units. The picture on
the bottom left shows unit folded then unfolded and is highlighted where the height is. The
picture on the bottom right shows the big cube and its base and height and where segment FI
could be. The formula for surface area ended up being 3(x)^2 and the formula for volume is
¼(x)^3√2.
Figure 23. Substitutions for x in Small Cube
Figure 23 shows 6 inches being substituted for x in the surface area and volume formulas.
X represents the side length of a piece of origami paper, and the origami paper used is 6 inches
long. The actual Surface Area of the small cube is 27in.^2 and the actual Volume is 6.75√2in.^3.
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Figure 24. Substitutions for x in Big Cube
Figure 24 shows 6 inches being substituted for x in the surface area and volume formulas.
X represents the side length of a piece of origami paper the origami paper used was 6 inches
long. The actual surface area of the big cube is 108in.^2 and the actual volume is 54√2in.^3.
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Figure 25. Total Surface Area of Model and Substitution for x
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Figure 25 shows the process for finding the surface area of the whole model including the
small cube and the big cube excluding the gaps in it. The surface area formula of the cube
including the gaps is 3(x)^2 and since the area of the small cubes fills all the gaps of the big cube
the whole model’s surface area is also 3(x)^2. When 6 inches is substituted for x the surface area
becomes 108in.^2
Figure 26. Deriving the Formula for the Empty Space and Substitution for x
Figure 26 shows how to get the formula for the empty space by subtracting the volume of
the small cube from the volume of the big cube. The empty space is the area between the small
cube and the big cube. The formula for the empty space is 7/32(x)^3√2 and when the side length
of 6 inches is substituted for x the volume of the empty space is 47.25√2in.^3. Another way to
find the volume of the empty space, not shown in Figure 5, is by first noticing that the volume of
the small cube is 1/8 of the volume of the big cube. From there you can infer the eight small
cubes make up the volume of the big cube and if you subtract one small cube then you get the
volume of seven small cubes. This would look like 7*1/32(x)^3√2 which is really 7/32(x)^3√2,
the formula for the empty space that was derived from Figure 5.
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In conclusion, cubes are very intriguing. For example from Figure 5 we can find the
volume of the empty space by subtracting of the small cube from the big cube or just multiply
the small cube by seven. In a similar way, it can be found impressive that the volume and surface
area of cubes are just two simple formulas, side^3, and 6(base) respectively. Also, one might
view cubes as an interesting subject because they can easily be made in origami. All these
reasons and many more make cubes a very interesting topic not just in geometry but in every
field that they appear. Most importantly it was proven in each proof that it was possible to find a
side length of a model and then to use the cube formulas to figure out the cubes’ volume and
surface area. All these reasons and many more make cubes a very interesting topic not just in
geometry but in every field that they appear.