Important Formulas: Cubes & Cuboids | 3 Months Preparation for CAT PDF Download

Cube

A cube is a three-dimensional geometric figure characterized by six identical square faces, right angles, and equal edge lengths. A cuboid with equal measurements in length, breadth, and height is referred to as a cube.
A cube is a solid object enclosed by six square planar surfaces, with the length of its sides referred to as edges.

Formulas for Cube

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 1 sides painted = 6(n-2)³
• Number of cubes with 2 sides painted = 12(n-2)
• Number of cubes with 3 sides painted= 8(always)

Cuboid 

A cuboid, commonly known as a rectangular prism, is a three-dimensional object featuring six rectangular faces, right angles, and edges of different lengths.
Notably, all edges of the cuboid are unequal, indicating that the measurements of length, breadth, and height differ.
A cuboid comprises six rectangles, and each of these rectangles is referred to as a face.

Formulas for Cuboid 

• For a cuboid of dimension a x b x c painted on all sides which is cut into smaller cubes of dimension 1 x 1 x 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 sides painted = 8

Instructions for Solving Formulas Related to Cubes and Cuboids 

A cube is cut in two equal parts along a plane parallel to one of its faces. One piece is then coloured red on the two larger faces and green on the remaining, while the other is coloured green on two smaller adjacent faces and red on the remaining. Each is then cut into 32 cubes of same size and mixed up.

Q1: How many cubes have only one coloured face each?
(a) 32
(b) 8
(c) 16
(d) 0
Ans: (c)
Sol: 
8 from (I) and 8 from (II)
Therefore 8 from each.

Q2: Choose the box that is similar to the box formed from the given sheet of paper (X).
Problem Figures

Answer Figures
(a) Only 1
(b) Only 2
(c) 1,2,3
(d) 1,3 and 4
Ans: (b)
Sol: When the sheet is folded, we find A lies opposite to D, B lies opposite to F, and C lies opposite to E. Hence, only option 2 is correct.