By the Pythagorean Theorem, 

The surface area of a cube having 6 sides, is 6 times the area of one of its sides. 

The area of any side of a cube is the square of the side length.

So if the side length is , the area of any side is , or .   

Thus the surface area of the cube is

Since the dice are  on each of their sides,  of them would measure  in length when standing face-to-face. This means  dice will fit along the edge of the box that measures .

 dice will also fit on the edge of the box measuring , but  will not fit since adding an additional die would bring the length of the dice standing face-to-face up to . There are no half-dice to fill the gap, so there is a small empty space on this side.

 dice will fit along the side that measures . 

Treating "dice" as the unit of measurement instead of  and considering only the volume where the dice will fit, one ends up with a rectangular shape measuring  dice   dice   dice. The volume of this shape is the number of dice that will fit in this area (and thus the box). Find the volume by multiplying all lengths of the shape's sides together:

 dice  dice. 

When searching for the volume of a cube we are looking for the amount of the space enclosed by the cube.

To find this we must know the formula for the volume of a cube which is 

Using this formula we plug in the side length for  to get 

Cube the side length to arrive at the answer of 

 The answer is .

To find this we must know the formula for the volume of a cube which is

We plug the volume into the equation yielding 

Then we take the cube root of both sides giving us 

We now know 

The answer is .

Vertices = three planes coming together at a point = 8

Edges = two planes coming together to form a line = 12

Faces = one plane as the surface of the solid = 6

Vertices + Edges + Faces = 8 + 12 + 6 = 26

In order to solve this problem, we need to make sure that the measurements given are in uniform units. In this case, we will use centimeters; therefore, we need to convert the other units of meters and millimeters into centimeters via dimensional analysis.

Solve for the width.

Solve for the height.

Now, solve for volume by using the formula .

If you calculated , then you did not convert into appropriate units.

A few facts need to be known to solve this problem. Observe that the diagonal of the square face of the cube cuts the qradilateral into two right isosceles triangles; therefore, the length of a side of the square to its diagonal is the same as an isosceles right triangle's leg to its hypotenuse: .

Rearrange an solve for .

Now, solve for the volume.

In order to solve this question, we need to convert miles to meters via dimensional analysis and then solve for the volume of the building.

Now, solve for volume.

The volume of a cube is one side cubed.  Because we know that one face has an area of 16 in², then we know that one side must be the square root of 16 or 4.  Thus the volume is .

We only have the length of one edge, but that is sufficient for finding the volume.  The volume of a cube equals one edge multiplied by itself three times: