The first thing necessary is to determine the dimensions of the cube. This can be done using the surface area formula for cubes: A = 6s², where s is the length of the cube. For our data, this is:

6s² = 294

s² = 49

(taking the square root of both sides) s = 7

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (7,7,7). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):

d = √((x₁ – x₂)² + (y₁ – y₂)²  + (z₁ – z₂)²) 

Or for our simpler case:

d = √((x)² + (y)²  + (z)²) = √( (s)² + (s)²  + (s)²) = √( (7)² + (7)²  + (7)²) = √( 49 + 49 + 49) =  √(49 * 3) = 7√(3)

The volume of a rectangular prism is .

We are told that the shortest edge is 3.  Let us call this the height. 

We now have , or .

Now we replace variables by known values:

Now we have:

We have thus determined that the other two edges of the rectangular prism will be 4 and 12.  We now need to find the longest diagonal.  This is equal to:

If you do not remember how to find this directly, you can also do it in steps.  You first find the diagonal across one of the sides (in the plane), by using the Pythagorean Theorem.  For example, we choose the side with edges 3 and 4.  This diagonal will be:

We then use a plane with one side given by the diagonal we just found (length 5) and the other given by the distance of the 3rd edge (length 12). 

This diagonal is then .

The number of square units in the surface area of a cube is given by the formula 6s², where s is the length of the side of the cube in units. Moreover, the number of cubic units in a cube's volume is equal to s³. 

Since the number of square units in the surface area is twice as large as the cubic units of the volume, we can write the following equation to solve for s:

6s² = 2s³

Subtract 6s² from both sides.

2s³ – 6s² = 0

Factor out 2s² from both terms.

2s²(s – 3) = 0

We must set each factor equal to zero. 

2s² = 0, only if s = 0; however, no cube has a side length of zero, so s can't be zero.

Set the other factor, s – 3, equal to zero.

s – 3 = 0

Add three to both sides.

s = 3

This means that the side length of the cube is 3 units. The volume, which we previously stated was equal to s³, must then be 3³, or 27 cubic units.

The answer is 27. 

In order to find the surface area of a cube, we need to solve for the length of each side, .

Recall the formula for volume:

Plug in what we know and solve for :

Now plug this value into the surface area formula:

The volume of a cube is length x width x height. Since it's a cube, though, the length, width, and height are all equal, and equivalent to the length of one edge of the cube.  Therefore, to find the lenght of an edge of the cube, just find the cube root of the volume. In this case, the cube root of 512 is equal to 8.

All the edges of a cube have the same length, and the volume of a cube is the length of an edge taken to the third power.

So if we take the edge of the cube to be of length x, then:

So the length of the edge of a cube with a volume of 125 is 5.

The formula for the volume of a cube is 

where  is the length of a side.

Here, the volume is 729. To find the side length, take the cube root of both sides:

The cube root of 729 is 9, so the length of each side of the cube is 9 feet.

The formula for the volume of a cube is  where s is any edge.

This means one edge of the cube is . We then divide 8.5 by 12 to convert to feet.

 feet.

The area of an equilateral triangle is given by the formula

Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is 

Substitute :

 square inches.