Cube and Cuboid

Intrduction:

• In a cube or a cuboid there are six faces in each.
• In a cube length, breadth and height are same while in cuboid these are different.
• In a cube the number of unit cubes = (side)³.
• In cuboid the number of unit cube = (l x b x h).

Example:

A cube of each side 4 cm, has been painted black, red and green on pars of opposite faces. It is then cut into small cubes of each side 1 cm.

The following questions and answers are based on the information give above:

1. How many small cubes will be there ?

Total no. of cubes = (sides)³ = (4)³ = 64

2. How many small cubes will have three faces painted ?

From the figure it is clear that the small cube having three faces coloured are situated at the corners of the big cube because at these corners only three faces of the big cube meet.

Therefore the required number of such cubes is always 8, because there are 8 corners.

3. How many small cubes will have only two faces painted ?

From the figure it is clear that to each edge of the big cube 4 small cubes are connected and two out of them are situated at the corners of the big cube which have all three faces painted.

Thus, to edge two small cubes are left which have two faces painted. As the total no. of edges in a cube are 12.

Hence the no. of small cubes with two faces coloured = 12 x 2 = 24

(or)

No. of small cubes with two faces coloured = (x - 2) x No. of edges

where x = (side of big cube / side of small cube)

4. How many small cubes will have only one face painted ?

The cubes which are painted on one face only are the cubes at the centre of each face of the big cube.

Since there are 6 faces in the big cube and each of the face of big cube there will be four small cubes.

Hence, in all there will be 6 x 4 = 24 such small cubes (or) (x - 2)² x 6.

5. How many small cubes will have no faces painted ?

No. of small cubes will have no faces painted = No. of such small cubes

= (x - 2)³ [ Here x = (4/1) = 4 ]

= (4 - 2)³

= 8.

6. How many small cubes will have only two faces painted in black and green and all other faces unpainted ?

There are 4 small cubes in layer II and 4 small cubes in layer III which have two faces painted green and black.

Required no. of such small cubes = 4 + 4 = 8.

7. How many small cubes will have only two faces painted green and red ?

No. of small cubes having two faces painted green and red = 4 + 4 = 8.

8. How many small cubes will have only two faces painted black and red ?

No. of small cubes having two faces painted black and red = 4 + 4 = 8.

9. How many small cubes will have only black painted ?

No. of small cubes having only black paint. There will be 8 small cubes which have only black paint. Four cubes will be form one side and 4 from the opposite side.

10. How many small cubes will be only red painted ?

No. of small cubes having only red paint = 4 + 4 = 8.

11. How many small cubes will be only green painted ?

No. of small cubes having only green paint = 4 + 4 = 8.

12. How many small cubes will have at least one face painted ?

No. of small cubes having at least one face painted = No. of small cubes having 1 face painted + 2 faces painted + 3 faces painted

= 24 + 24 + 8

= 56.

13. How many small cubes will have at least two faces painted ?

No. of small cubes having at least two faces painted = No. of small cubes having two faces painted + 3 faces painted

= 24 + 8

= 32.

Directions to Solve The following questions are based on the information given below: A cuboid shaped wooden block has 6 cm length, 4 cm breadth and 1 cm height. Two faces measuring 4 cm x 1 cm are coloured in black. Two faces measuring 6 cm x 1 cm are coloured in red. Two faces measuring 6 cm x 4 cm are coloured in green. The block is divided into 6 equal cubes of side 1 cm (from 6 cm side), 4 equal cubes of side 1 cm(from 4 cm side).

1.  How many cubes having red, green and black colours on at least one side of the cube will be formed ? A. 16 B. 12 C. 10 D. 4 Answer & Explanation 2.  How many small cubes will be formed ? A. 6 B. 12 C. 16 D. 24 Answer & Explanation 3.  How many cubes will have 4 coloured sides and two non-coloured sides ? A. 8 B. 4 C. 16 D. 10 Answer & Explanation 4.  How many cubes will have green colour on two sides and rest of the four sides having no colour ? A. 12 B. 10 C. 8 D. 4 Answer & Explanation 5.  How many cubes will remain if the cubes having black and green coloured are removed ? A. 4 B. 8 C. 12 D. 16 Answer & Explanation

Directions to Solve The following questions are based on the information given below: There is a cuboid whose dimensions are 4 x 3 x 3 cm. The opposite faces of dimensions 4 x 3 are coloured yellow. The opposite faces of other dimensions 4 x 3 are coloured red. The opposite faces of dimensions 3 x 3 are coloured green. Now the cuboid is cut into small cubes of side 1 cm.

1.  How many small cubes will have only two faces coloured ? A. 12 B. 24 C. 16 D. 12 Answer & Explanation 2.  How many small cubes have three faces coloured ? A. 24 B. 20 C. 16 D. 8 Answer & Explanation 3.  How many small cubes will have no face coloured ? A. 1 B. 2 C. 4 D. 8 Answer & Explanation 4.  How many small cubes will have only one face coloured ? A. 10 B. 12 C. 14 D. 18 Answer & Explanation

Directions to Solve The following questions are based on the information given below: A cuboid shaped wooden block has 4 cm length, 3 cm breadth and 5 cm height. Two sides measuring 5 cm x 4 cm are coloured in red. Two faces measuring 4 cm x 3 cm are coloured in blue. Two faces measuring 5 cm x 3 cm are coloured in green. Now the block is divided into small cubes of side 1 cm each.

1.  How many small cubes will have will have three faces coloured ? A. 14 B. 8 C. 10 D. 12 Answer & Explanation 2.  How many small cubes will have only one face coloured ? A. 12 B. 28 C. 22 D. 16 Answer & Explanation 3.  How many small cubes will have no faces coloured ? A. None B. 2 C. 4 D. 6 Answer & Explanation 4.  How many small cubes will have two faces coloured with red and green colours ? A. 12 B. 8 C. 16 D. 20 Answer & Explanation

Directions to Solve The following questions are based on the information given below: All the faces of cubes are painted with red colour. The cubes is cut into 64 equal small cubes.

1.  How many small cubes have only one face coloured ? A. 4 B. 8 C. 16 D. 24 Answer & Explanation 2.  How many small cubes have no faces coloured ? A. 24 B. 8 C. 16 D. 0 Answer & Explanation 3.  How many small cubes are there whose three faces are coloured ? A. 4 B. 8 C. 16 D. 24 Answer & Explanation 4.  How many small cubes are there whose two adjacent faces are coloured red ? A. 0 B. 8 C. 16 D. 24 Answer & Explanation

Directions to Solve The following questions are based on the information given below: All the opposite faces of a big cube are coloured with red, black and green colours. After that is cut into 64 small equal cubes.

1.  How many small cubes are there where one face is green and other one is either black or red ? A. 28 B. 8 C. 16 D. 24 Answer & Explanation 2.  How many small cubes are there whose no faces are coloured ? A. 0 B. 4 C. 8 D. 16 Answer & Explanation 3.  How many small cubes are there whose 3 faces are coloured ? A. 4 B. 8 C. 16 D. 24 Answer & Explanation 4.  How many small cubes are there whose only one face is coloured ? A. 32 B. 8 C. 16 D. 24 Answer & Explanation 5.  How many small cubes are there whose at the most two faces are coloured ? A. 48 B. 56 C. 28 D. 24 Answer & Explanation

Directions to Solve All the faces of a cube are painted with blue colour. Then it is cut into 125 small equal cubes.

1.  How many small cubes will be formed having only one face coloured ? A. 54 B. 8 C. 16 D. 24 Answer & Explanation 2.  How many small cubes will be formed having no face coloured ? A. 27 B. 8 C. 16 D. 24 Answer & Explanation

Directions to Solve All the six faces of a cube of a cube are coloured with six different colours - black, brown, green, red, white and blue. Red face is opposite to the black face. Green face is between red and black faces. Blue face is adjacent to white face. Brown face is adjacent to blue face. Red face is in the bottom.

1.  The upper face is _________ A. White B. Black C. Brown D. None of these Answer & Explanation 2.  The face opposite to brown is _________ A. Blue B. White C. Green D. Red Answer & Explanation 3.  Which of the following is adjacent to green ? A. Black, white, brown, red B. Blue, black, red, white C. Red, black, blue, white D. None of these Answer & Explanation 4.  Which face is opposite to green ? A. Red B. White C. Blue D. Brown Answer & Explanation

Directions to Solve There are 128 cubes with me which are coloured according to two schemes viz. 64 cubes each having two red adjacent faces and one yellow and other blue on their opposite faces while green on the rest. 64 cubes each having two adjacent blue faces and one red and other green on their opposite faces, while red on the rest. They are then mixed up.

1.  How many cubes have at least two coloured red faces each ? A. 0 B. 32 C. 64 D. 128 Answer & Explanation 2.  What is the total number of red faces ? A. 0 B. 64 C. 320 D. 128 Answer & Explanation 3.  How many cubes have two adjacent blue faces each ? A. 64 B. 32 C. 0 D. 128 Answer & Explanation 4.  How many cubes have only one red face each ? A. 128 B. 32 C. 64 D. None Answer & Explanation 5.  Which two colours have the same number of faces ? A. Red and Yellow B. Blue and Green C. Red and Green D. Red and Blue Answer & Explanation

Directions to Solve 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.

1.  How many cubes have only one coloured face each ? A. 32 B. 8 C. 16 D. 0 Answer & Explanation 2.  What is the number of cubes with at least one green face each ? A. 36 B. 32 C. 38 D. 48 Answer & Explanation 3.  How many cubes have two red and one green face on each ? A. 0 B. 8 C. 16 D. 4 Answer & Explanation 4.  How many cubes have no coloured face at all ? A. 32 B. 8 C. 16 D. None Answer & Explanation 5.  How many cubes have each one red and another green ? A. 0 B. 8 C. 16 D. 24 Answer & Explanation