A cube of side 125 cm is painted red on all the faces and then cut into smaller cubes of sides 25 cm each. Find the number of smaller cubes having at least two faces painted.

Given:

Length of the side of the big cube = 125 cm.

Length of the side of the smaller cube = 25 cm.

Formula:

A cube is a three-dimensional figure with 8 corners, 6 surfaces, and 12 edges (sides).

If a cube is painted on all of its surfaces with the same colour, and then divided into smaller cubes of equal size, then the number of smaller cubes so obtained will be calculated as follows:

• Number of cubes with no face painted = (n - 2)³
• Number of cubes with one face painted = 6(n - 2)²
• Number of cubes with two faces painted = 12(n - 2)
• Number of cubes with three faces painted = 8(Always)

 where n = No. of parts of each side = Length of the side of larger cube ÷ length of the side of the smaller cube

Calculation:

n = 125 ÷ 25 = 5

⇒ n - 2 = 3

Number of smaller cubes having at least two faces painted

= total no. of cubes - (no. of cubes with no face painted + no. of cubes with only one face painted)

= 125 - (3³ + 6 × 3²)

= 125 - (27 + 54)

= 125 - 81

= 44

Hence, 44 is the correct answer.