- Total Number of Faces = 6xyz (width (x), length (y), and height (z)).
In regard to finding how we can obtain the ‘total number of hidden faces’ the following rule can be applied:
Formula Three
- Hidden Faces = Total Number of Faces – Total Number of Visible Faces
By using this formula we can find the total number of hidden faces on a/many cube, by subtracting the total number of faces from the total number of visible faces, which will give us the total number of hidden faces.
To check whether this equation is correct we can add the total number of hidden and seen faces together so that it can give us the total number of faces.
Using the information that I have researched I am now going to find the total number of hidden faces for when 8 cubes are joined together in a row.
Representing visual faces:
-
8 x 6 = 48 (formula 1)
- 48 – 26 = 22 (formula 2)
- (Check the answer) 22 +26 = 48
Below is a table (Table 1.1) showing the total number cubes, total number of hidden and visible faces in rows.
<Data obtained by using diagrams in appendix one>
Formula Three in theory is not much efficient and can be time consuming. It will be very difficult and time consuming for example for a person who needs to find the total number of hidden faces for 35 cubes in a row. They would need to draw or build the cubes, count the number of visible faces and then use formula three to find the number of hidden faces.
Therefore I am going to try and find a pattern between the ‘number of cubes’ and the ‘number of hidden faces’, by using the data in table 1.1.
Let n be the ‘n’ be the number of cubes:
Experimenting to find a formula for hidden faces for (a) cube(s).
Try 2n – 2
(2 * 1) – 2 = 0 formula does not work
Try 2n + 2
(2 * 1) + 2 = 4 formula does not work
Try 3n + 2
(3 * 1) + 2 = 5 formula does not work, however the answer given is the same value
for the visible faces (referring to table 1.1)
Try for more number of cubes
Let n = 2
(3*2) + 2 = 8 correct
Let n = 3
(3*3) + 2 = 11 correct
Let n = 7
(3*7) + 2 = 21 correct
The following formula ‘3n + 2’ can be used to find the total number of visible faces.
Back to finding the formula for hidden faces, if 3n + 2 is used to find the number of visible faces, lets try 3n – 2 to find the number of hidden faces.
Let n = 1
(3*1) – 2 = 1 correct
Let n = 2
(3*2) – 2 = 4 correct
Let n = 3
(3*3) – 2 = 7 correct
Let n = 4
(3*4) – 2 = 10 correct
It seems as though the formula does work, from Table 1.1, I am going to choose the number of cubes as 11 and use the formula against this number of cubes. To check that formula is correct to use to find the number of hidden faces on any cube(s).
Let n = 11
(3*11) – 2 = 31 correct. Formula does work!
Fundamentally the above working outs show 2 formulas:
Formula 1A: Total number of visible faces = 3n + 2
Formula 1B: Total number of hidden faces = 3n – 2
Task 2
When a cuboid is formed, cubes are joined in different ways i.e. in rows and columns. In Task 1 we found out that the total number of faces of a/many cubes can be found in 2 different ways (formula one and formula two).
I am now going to investigate the number of hidden faces in a cuboid.
Below is a cuboid made from 30 cubes; fig 1.
X, Y and Z represent width, length and height.
Using ‘6(xyz)’ will give me the total number of faces (as from task one we already know that total number of cubes multiplied by 6 equals the total number of faces).
The visible faces of the cuboid can be found by using the following formula:
(2yz + 2xz + xy)
2yz – by multiplying y and z we are given the number of faces on one side of the cuboid (the length side) multiplying it by 2 gives us the number of faces on the other side too. Thus the expression gives the total number of faces on both sides of the cuboid (length side).
2xz – this expression gives the total number of faces on both sides of the cuboid (width side).
xy – gives the total number of faces on top of the cuboid. We do not multiply this expression by 2 as the bottom part of the cuboid is of hidden faces.
Thus
Formula A, for total number of faces : 6xyz
Formula B, for total number of visible faces: (2yz + 2xz + xy)
Using the rule that I mentioned in Task1;
Formula Three; Hidden faces = Total number of faces – Total number of visible faces.
The formula for total number of hidden faces in a cuboid is:
Formula C: 6xyz – (2yz + 2xz + xy)
Using figure 1 (30 cubes: cuboid) I am going to put this formula to test.
x = 3, y = 5, z = 2
Total number of hidden faces = (6x3x5x2) - (2x5x2 + 2x3x2 + 3x5)
= 180 – ( 20 + 12 + 15)
= 180 – 47
= 133
<Data obtained by using diagrams in appendix two>
The following are sequences that I found in the above table:
Number of cubes in a cuboid: 12, 18, 24, 30, 36….. value + 6 each time.
Number of visible faces: 26, 33, 40, 47, 54….. value + 7 each time.
Number of hidden faces: 46, 75, 104, 133, 162…. value + 29 each time
Sequence found between number of cubes in a cuboid and the number of visible faces:
12 – 26 +14
18 – 33 +15
24 – 40 +16
30 – 47 +17
36 – 54 +18
These are the patterns that I have obtained from Table 1.2, however in regard to my formula for the total number of faces, visible faces and hidden faces I do not see any use of them being used.
Task 2 was accomplished as well successful by using formulas/rules that were applied in Task 1. I basically used formulas from Task 1 as a base to produce formulas in Task 2.
Counting either in a drawing or on a model for faces on a cube(s)/cuboid can be very time consuming. My report gives all the necessary formulas that can be used by anyone for finding out the total number of faces, visible and hidden on any cube(s)/cuboids.
