# Formulas For Cube And Cuboid

## Cuboid :

A Cuboid is a closed 3-dimensional geometrical figure bounded by six rectangular plane regions.

A Cuboid is made up of six rectangles, each of the rectangle is called the face. In the figure above, ABFE, DAEH, DCGH, CBFG, ABCD and EFGH are the 6-faces of cuboid. The top face ABCD and bottom face EFGH form a pair of opposite faces. Similarly, ABFE, DCGH, and DAEH, CBFG are pairs of opposite faces. ## Cube:

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.

#### Formulas for Cube  :

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

## Directions to Solve Cube and Cuboid Formulas

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.

### Question 1.

How many cubes have only one coloured face each ?

(A)  32
(B)  8
(C) 16
(D) 0

#### Explanation: 8 from (I) and 8 from (II)

Therefore 8 from each.

### Question 2.

What is the number of cubes with at least one green face each ?

(A) 36
(B) 32
(C) 48
(D) 38

#### Explanation: 24 from (I) and 14 from (II)

### Question 3.

How many cubes have each one red and another green ?

(A) 0
(B) 8
(C) 16
(D) 24  