Let's understand Cubes | Hamaraguru

What is a cube? In Geometry, a cube is a 3-dimensional solid object bounded by 6 square faces, facets or sides, with three meetings at each vertex. It has 12 edges and 8 corners.

Let's understand Cubes | Hamaraguru

What is a cube?
In Geometry, a cube is a 3-dimensional solid object bounded by 6 square faces, facets or sides, with three meetings at each vertex. It has 12 edges and 8 corners.
Important points to remember:

1. If a cube is cut into (a+b+c) cuts such that 'a' cuts on x-axis, 'b' cuts on y-axis, 'c' cuts on z-axis, then the total number of cubes / cuboids (identical / non-identical) the cube can be divided into is (a+1)x(b+1)x(c+1).

2. A cube can be cut in x, y, z-axis, to get a maximum number of identical pieces, we need to cut the cube in all the three axes such that the difference between one axis cut and the other axis cut must be minimum.

​​​In other words, the cuts should be uniformly distributed as possible among three axes such that the number of cuts made in at least two of the axes is equal. (If not all the three)

Example 1:

What is the maximum number of identical pieces we obtain if a large cube is cut 13 times?

Sol:

13 = 5 + 4+ 4

We will get the maximum number of pieces when a cube is cut into 3 axes through 5, 4, 4 cuts.

Now each cut will give you an additional identical piece.

Hence the number of pieces = (5+1)(4+1)(4+1) = 150

Example 2:

What is the maximum number of identical pieces we obtain if a large cube is cut 11 times?

Sol:

11 = 4 + 4 +3

We will get the maximum number of pieces when a cube is cut into 3 axes through 3, 4, 4 cuts.

Now each cut will give you an additional identical piece.

Hence the number of pieces = (3+1)(4+1)(4+1) = 100

 

3. When a cube is painted with one color and cut into identical smaller cubes:

a) Total number of cubes = 

b) One face painted cubes = 

c) Two faces painted cubes = 12 (n-2)

d) Three faces painted cubes = 8

e) None of the faces colored = , where n is the number of pies on each side.

Cube 

(nxn)

No of Cubes   

0 side painted

1 side painted

2 sides painted

12(n-2)

3 sides painted

8

2x2

 8

0

0

0

8

3x3

27

1

6

12

8

4x4

64

8

24

24

8

5x5

125

27

54

36

8

6x6

216

64

96

48

8

7x7

343

125

150

60

8

8x8

512

216

216

72

8

9x9

729

343

294

84

8

10x10

1000

512

384

96

8

Example 3:

 

A. A large cube is painted with red and cut into 125 identical small cubes. What is the total number of small cubes that has only one face painted?

Sol:

The number of cuts that were made uniform along the three axes is 5.

Formula: One face painted cubes =  = 6(9) = 54.

 

B. What is the total number of small cubes that has three faces painted?

Sol:

Formula: 3 faces painted cubes = 8.

So, the answer is 8.

 

C. What is the total number of small cubes without having red paint?

Sol:

Formula: None of the faces colored =

We know that n=5.

So, the answer is = 27