Cubes and Cuboids

Question 1

How many faces are on a cube or cuboid?

Answer 1

6 faces

Question 2

What’s the formula for the number of unit cubes in a cube?

Answer 2

(side)³

Question 3

What’s the relationship between the length, breadth and height in a cube?

Answer 3

The length, breadth and height are all the same

NOTE: In a cuboid, the length, breadth, and height are all different values.

Question 4

What’s the formula for the number of unit cubes in a cuboid?

Answer 4

l * b * h, where l is length, b is breadth, and h is height

Question 5

What is the maximum number of faces on a cube or cuboid that you can see at any angle?

Answer 5

3 faces; there are 6 faces, 3 pairs of opposite faces, and you cannot see a pair of opposite faces at the same angle

Question 6

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, what is the formula for the number of cubes with none of the sides painted?

Answer 6

(n - 2)³

Question 7

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, what is the formula for the number of cubes with only 1 side painted?

Answer 7

6(n - 2)²

Question 8

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, what is the formula for the number of cubes with 2 sides painted?

Answer 8

12(n - 2)

Question 9

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, how many cubes have 3 sides painted?

Answer 9

8

Question 10

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, what is the formula for the number of cubes with none of the sides painted?

Answer 10

(a - 2)(b - 2)(c - 2)

Question 11

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, what is the formula for the number of cubes with only 1 side painted?

Answer 11

2[(a - 2)(b - 2) + (b - 2)(c - 2) + (a - 2)(c - 2)]

Question 12

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, what is the formula for the number of cubes with 2 sides painted?

Answer 12

4(a - 2) + 4(b - 2) + 4(c - 2) = 4(a + b + c -6)

Question 13

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, how many cubes have 3 sides painted?

Answer 13

8

Question 14

What is the formula for the maximum number of unit cubes you can see from any angle, for a cube with sides n x n x n?

Answer 14

(n - 2)² * 3 + 9 * (n - 2) + 7

Question 15

What is the formula for the maximum number of unit cubes you can see from any angle, for a cuboid with sides a x b x c?

Answer 15

3(a + b + c - 6) + [(a - 2)(b - 2) + (b - 2)(c - 2) + (a - 2)(c - 2)] + 7