To investigate the hidden faces and the number of faces seen on a cube or a cuboids when it’s placed on a table.

and these are the extra faces seen on the side.

3x-2= Hidden faces

Number of cubes multiplied by 3, I've done this because on each cube you can see 3 faces including the middle cubes, we than minus 2 because there's 2 extra faces seen at the ends, we took this as though we couldn't see it. So the formula is 3x-2, this is opposite to the number of faces. I tried other formulas at first but none of them worked and so I found different which worked.

2)

Number of cubes (x)

Hidden faces

Number of faces seen

Total faces

5

6

2

3

9

2

3

5

3

8

4

7

7

24

5

9

21

30

0

9

41

60

Number of faces seen= 4x+1

Number of cubes multiplied

by 4, we done this because on

each cube you can see 4 faces

and than we add 1 because

you can see an extra face at

the top.

We put the results in a table and found out that the hidden faces went up by 2 when we increased the number of cubes. The 1st investigation went up by 3 and so the 1st ones formula was multiplied by 3 so I thought this one was multiplied by 2 and I minus 1 because there was only one side which we could see.

3)

Number of cubes (x)

Hidden faces

Number of faces seen

Total faces

4

2

2

24

6

20

6

36

8

28

20

48

4x-4=Hidden faces

You multiply it by 4 because you can see 4 faces, I minus 4 faces because you can see 4 sides but I took this as if I couldn't see.

2x+4=number of faces seen.

I multiplied it by 2 because the numbers of cubes are going in 2's so we add 4 because you can see 4 sides.

4)

Number of cubes (x)

Hidden faces

Number of faces seen

Total faces

4

2

2

24

8

28

20

48

2

44

28

72

6

60

36

96

4x-4=Hidden faces

You multiply it by 4 because you can see 4 faces, I minus 4 faces because you can see 4 sides but I took this as if I couldn't see.

2x+4=number of faces seen.

I multiplied it by 2 because the numbers of cubes are going in 2's so we add 4 because you can see 4 sides.

*If you find the volume than you can times the volume by 6 which will equal to something E.g. 20×6=120

Than you count the faces you can see and minus the faces you can see from 120 which will give you the number of hidden faces.

I have found out the overall formula for a cube and cuboids because it was linked to volume and surface area.

The overall formula for the number of faces seen is, L×W+(H×L)×2+(H×W)×2

And the formula to find out the hidden faces is,

L×W×H×6-Number of faces seen.

Number of faces seen= (L×W) + (H×L) ×2+ (H×W) ×2

Hidden faces= L×W×H×6-Number of faces seen.

Here are some examples to prove that my formula works.

This formula works with a cubes and cuboids.

There is only one variable, because if we had more than 1 variable than it would be hard to find out the link between each variable.

If I was to carry on I would change the shapes in to a different shape or to investigate the different patterns that occur with different cubes and cuboids when all the faces are painted of a large cube or a cuboids and then that is separated into smaller cubes and then how many faces of each small cube are still painted.