c. 4 x 4 x 4 Cube
- Explanation of why there isn’t a length of 1
There cannot be a length 1 as then this would obviously just be 1 at every measurement .It would also only have 6 faces
- Predictions of a 6 x 6 x 6 cube.
Once I had half filled in my chart , with the 2,3,4 chart I was quite sure that I could see a few patterns within the data. First of all the most noticeable was the 2 faces, which was running in the 12 times tables, the totals were quite easy to see that it was in fact the length of cube squared. I had also worked out that 3 faces would always be 8 as any cube has 8 corners.
I was very confident that I had worked out all of the formulas by now. Using the formula I had created I filled in the table as below.
LENGTH OF CUBE
The formula I had at this point created was as follows.
5. Explanation of the 10 x 10 x 10 cube
This was relatively easy to work out, the total was1000 cubes, because it was 10 x 10 x 10.
The zero face was accomplished by taking the 2 adjacent cubes away then cubed .The actual sum was 10-2 = 8 x 8 x8 = 512.
The one face was worked out by subtracting the 2 corner cubes from each length of 10 (N) then multiplying by 6. The actual sum was 10 – 2 = 8 x 8 = 64 x 6 = 384.
The two face sum was worked out by 10 -2 =8 x 12 = 96
The 3 face corners will always be 8
6. Formula Explanation
The first formula that I worked out was the total no of cubes, this was simple as to measure the volume of a cube is to cube the length that you have. This equaled to
n x n x n = n 3.
The second formula was for 0 faces, the formula for this was (n – 2) 3. The -2 is because there are 2 cubes adjacent to every side of the internal cubes , then it would be cubed minus the outer edged each side of the cubes, as it is also a cube albeit a smaller one.
The third formula was for one red face, the formula for this was 6[ ( n-2) 2 ]. As in the previous formula it would be -2 as it would have 2 adjacent cubes that are 1 on each side of the outer edge, so it stands to reason that it would be – 2, then it was squared as that is the area of each flat face with one each side, from there it was multiplied by 6 as that is how many sides are on a cube.
The forth formula was for 3 faces, this would be 8 as a cube has 8 corners, the number would always be 8 no matter how big the cube
For the last formula to work out 2 faces. I first worked this out in a logical but extended way. As I had already worked out the formula for the total amount of cubes, 0 faces, ,1 face and 3 faced corner cubes it made sense to take the total formula and minus all the other formula from it. The formula for this was n3 – (n-2)3 – 6[(n-2)2 – 8. I then realized there should be a much simpler formula. Once I took another look at the table I realized it would be 12(n-2).This was because there are 12 edges on a cube and as there are -1 at each edge( for the corners), this would mean it had to be n -2 .
Therefore I had to redo my table to show this correction. As shown below
7. The 10 x 10 x 10 table
Total
8. Introduction to Part 2 The Big Painted Red Cube
I was then given a worksheet to complete, which took things 1 step further to evaluate the differences for a general cuboid. I will try and show you how I did this in the next part.
9. How the cuboid works for the 5 x 4 x 3
10. Other examples of different cuboids
a. 4 x 5 x 7
b.6 x 4 x5
11. Formula Explanations
Formula table for a general cuboid.
The 0 face formula works because when you look at the cube you need to take the edges off every side of the cube to get to the middle. Hence Length – 2, Width -2 and Height – 2.
The 1 face formula works because yet again you have to take the edge blocks off each visible side of the cube, so if the front would be Length – 2 and Height – 2 then multiply the 2 together as you would to find the visible area, the same for other surfaces of the cuboid. As in a general cuboid you would have 3 different lengths as in L, W and H, then add them all together.
The 2 faces formula is a little bit different you need the edges of the Length, Width and Height, within a cuboid there are 4 widths, 4 lengths and 4 heights so therefore if you subtract the 2 corner cubes off each and then multiply each by 4,then add them all together.
The 3 face formula is the same as a cube as no matter what the size it will always have 8 corners no matter what size.
The total formula will be Width x Height x Length to get the volume.
12. Explanation of how a Cuboids Formula turns into the Cube Formula
The formula for o faces within a cuboid is ( L-2)(w-2)(h-2), if these were all the same length as in a cube the letters would become all the same for example N it would be (N-2)3 , this is the same formula as a cube
The formula for 1 face is 2[( w -2)(L -2)] + 2[(h –2)(L-2)] + 2[(h-2)(w-2)] ,if they were all changed to the same letter for example N it would be 2[( n -2)(n-2)] + 2[(n –2)(n-2)] + 2[(n-2)(n-2)], this would equalize to be the same as a cube ,the known formula is 6 [ (n – 2) 2].
The formula for 2 faces is 4(L-2) + 4( h-2) + 4(w-2), if these were all changed to the same letter , for example N it would look like this, 4(n-2) + 4( n-2) + 4(n-2), as in the previous formulas this also equalizes to the same as the cube formula for 2 faces which is 12( n-2).
If the formula for of total cubes is lwh within a cuboid if changed to the same letter for example n this would become n3, the same as the formula for a cube
