Recall that the volume of a cube is defined by the equation:

, where  is the side length of the cube. 

Therefore, if we know that , we can solve:

This means that .

Now, if we triple  to , the new volume of our cube will be:

Recall that the equation for the surface area of a cube is merely derived from the fact that the cube's faces are made up of  squares. It is therefore:

For our values, this is:

Solving for , we get:

, so 

Now, the volume of a cube is merely:

Therefore, for , this value is:

To find the volume of a cube, you multiple your side length 3 times (s*s*s).  

To find the side length from the volume, you find the cube root which gives you 4 

.  

Doubling the side gives you 8 

.  

The volume of the new cube would then be 512 

.

The surface area formula we need to solve this is 2ab + 2bc + 2ac.  So if we let a = 6, b = 3, and c = 2, then surface area:

= 2(6)(3) + 2(3)(2) +2(6)(2)

= 72 sq ft.

Surface area of a rectangular solid

= 2lw + 2lh + 2wh

= 2(6)(4) + 2(6)(3) + 2(4)(3)

= 108

Since the volume of the smaller cubes with edges, , is 27, we have:

.

The large cube has edges .

So the surface area of the large cube is:

.

The relationship can be determined, because it is possible to find the surface area of a cube from the volume and vice versa.

Quantity A:

To find the surface area of the cube, you must find the side length. To find the side length from the volume, you must find the cube root.  

Find the cube root of the volume.

Insert into surface area equation.

Quantity B:

To find the volume of a cube, you must find the side length. To find the side length from the surface area, you must divide by 6, then find the square root of the result. Then, cube that result.

Divide by 6.

Square root.

Now, to find the volume.

Quantity B is greater.

Begin by solving for the length of one side of the cube. Use the formula for surface area to do this:

s= length of one side of the cube

The length of the side of the cube is equal to the diameter of the sphere. Therefore, the radius of the sphere is 3. Now use the formula for the surface area of a sphere:

The surface area of the sphere is .

The surface area of a sphere is defined by the equation:

For our data, this means:

Solving for , we get:

 or 

The diameter of the sphere is .

Do not assume that the diameter will be half of the diameter of a sphere with volume of . Instead, begin with the sphere with a volume of . Such a simple action will prevent a vexing error!

Thus, we know from our equation for the volume of a sphere that:

Solving for , we get:

If you take the cube-root of both sides, you have:

First, you can factor out an :

Next, factor the :

Which simplifies to:

Thus, the diameter is double that or: