Formulas For Cube

Cubes Formulas & Definitions:-

A cuboid having its length, breadth, height all to be equal in measurement is called as a cube.
A cube is a solid bounded by six square plane regions, where the side of the cube is called edge.

 

For a cube of side n*n*n painted on all sides which is uniformly cut into smaller cubes of dimension 1*1*1,

  • Number of cubes with 0 side painted= (n-2) ^{3}
  • Number of cubes with 2 sides painted= 12(n-2)
  • Number of cubes with 1 sides painted =6(n – 2) ^{2}
  • Number of cubes with 3 sidess painted= 8(always)

 

 

Formulas for Cubes

Formulas for Cube

For a cuboid of dimension a*b*c painted on all sides which is cut into smaller cubes of dimension 1*1*1,

  • Number of cubes with 0 side painted= (a-2) (b-2) (c-2)
  • Number of cubes with 1 sides painted =2[(a-2) (b-2) + (b-2)(c-2) + (a-2)(c-2) ]
  • Number of cubes with 2 sides painted= 4(a+b+c -6)
  • Number of cubes with 3 sidess painted= 8

Example for Cube Formulas

Question 1: A cube having an edge of 12 cm each. It is painted red on two opposite faces, blue on one other pair of opposite faces, black on one more face and one face is left unpainted. Then it is cut into smaller cubes of 1 cm each.

The total no. of smaller cubes.

Solution: Total number of cubes= \frac{12\times 12\times 12}{1\times1\times 1} = 1728

Formula for cubes
Formula for cubes

Ques 2.The no. of smaller cubes which are having two-faces painted.

Solutions: For 2 sides painted, we look for the edges.

A cube has 12 edges;
8 edges, each edge having 10 cubes will have 2 sides painted. (4 edges of an unpainted side won’t be included).
We’ll also include those 4 cubes (which we didn’t count while counting 3 colored sides, as they have 2 sides painted)
Cubes on 4 edges of the unpainted side of the cube will have 1 side painted (due to the unpainted side).

Therefore, total cubes with 2 sides painted= 8*10 + 4= 84 cubes.