In this Page Formulas for Cube is given. 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 x n x n painted on all sides which is uniformly cut into smaller cubes of dimension 1 x 1 x 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}

Question 1: A cube having an edge of 12 cm each. It is painted green 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.Find the total no. of smaller cubes.

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

Question 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; in which 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 x 10 + 4 = 84 cubes.

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