The surface area of a rectangular prism with sides , , and  is given as:

. 

Two sides are known; it does not matter how they are designated, but for this problem let  and , with  as the unknown side. This yields equality:

Now that the three dimensions are known, it's possible to calculate the volume:

Recall that the equation for the volume of a cube is:

Since the sides of a cube are merely squares, the surface area equation is just  times the area of one of those squares:

So, for our two quantities:

Quantity A

Use your calculator to estimate this value (since you will not have a square root key). This is .

Quantity B

First divide by :

Therefore, 

Therefore, the two quantities are equal.

The surface area of a cube is made up of  squares. Therefore, the equation is merely  times the area of one of those squares.  Since the sides of a square are equal, this is:

, where  is the length of one side of the square.

For our data, we know:

This means that:

Now, while you will not have a calculator with a square root key, you do know that . (You can always use your calculator to test values like this.) Therefore, we know that . This is the length of one side

There are 6 sides to a cube. If the total surface area is 54 square inches, then each face must have an area of 9 square inches.

Every face of a cube is a square, so if the area is 9 square inches, each edge must be 3 inches.

First, we must ascertain the length of each side.  Based on our initial data, we know that the 6 faces of the cube will have a surface area of 6x². This yields the equation:

6x² = 486, which simplifies to: x² = 81; x = 9.

Therefore, each side has a length of 9.  Imagine the cube is centered on the origin.  This means its "front-left-bottom corner" will be at (–4.5, –4.5, 4.5) and its "rear-right-top corner" will be at (4.5, 4.5, –4.5).  To find the distance between these, we use the three-dimensional distance formula:

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

For our data, this will be:

√( (–4.5 – 4.5)² + (–4.5 – 4.5)² + (4.5 + 4.5)²) =

√( (–9)² + (–9)² + (9)²) = √(81 + 81 + 81) = √(243) =

√(3 * 81) = √(3) * √(81) = 9√(3)

The shortest length between any two non-adjacent corners will be the diagonal of the smallest face of the rectangular box. The smallest face of the rectangular box is a six-inch by six-inch square. The diagonal of a six-inch square is .

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 

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: