To begin, the best thing to do is to find the length of a side of the cube. This is done using the formula for the surface area of a cube. Recall that a cube is made up of  squares. Therefore, its surface area is:

, where  is the length of a side.

Therefore, for our data, we have:

Solving for , we get:

This means that 

Now, the diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or 

Now, if the the value of  is , we get simply 

The box is 2 times as long as it is high, so H = L/2. It is also 4 times as wide as it is long, so W = 4L. Now we need volume = L * W * H = L * 4L * L/2 = 2L³.

The surface area of a cube is merely the sum of the surface areas of the  squares that make up its faces. Therefore, the surface area equation understandably is:

, where  is the side length of any one side of the cube. For our values, we know:

Solving for , we get:

 or 

Now, the volume of a cube is defined by the simple equation:

For , this is:

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.