The volume of a rectangular prism is found using the following formula:

If we substitute our known values, then we can solve for the missing side.

Divide both sides of the equation by 12.

We now know that the missing length equals 7 centimeters.

This means that the box can have sides with the following dimensions: 3cm by 4cm; 7cm by 3cm; or 7cm by 4cm. The greatest area of one side belongs to the one that is 7cm by 4cm. 

To solve this problem we need an extension of the Pythagorean Theorem:

So the equation to solve becomes

So the distance of the diagonal is .

The length of the diagonal is from the bottom left hand corner closest to us to the top right hand corner that's farthest away from us. 

This kind of a problem may seem to be a little more complicated than it really is. 

In order to solve for the diagonal length, all that's required is the Pythagorean Theorem. This equation will be used twice to solve for the dashed line. 

For the first step of this problem, it's helpful to imagine a triangle "slice" that's being taken inside the prism. 

, where the diagonal of interest is D2, and D1 is the diagonal that cuts from corner to corner of the bottom face of the prism. Of this triangle that's outlined in pink dashed lines, the given information (the dimensions of the prism) provides a length for one of the legs (16). 

We can already "map out" that D2 (the hypotenuse of the dashed triangle) can be solved by using the Pythagorean Theorem if we can obtain the length of the other leg (D1).

The next step of this problem is to solve for D1. This will be the first use of the Pythagorean theorem. D1 is the diagonal of the base and is limited to a 2D face. This can be represented as:

The hypotenuse of the base, or the mystery length leg of the dashed triangle, can be solved by using the Pythagorean Theorem:

Now that we calculated the length of D1, D2 can be solved for by using the Pythagorean Theorem a second time:

To find the diagonal distance of a prism, you can use the formula: 

, where  = height;  = width, and  = length.  

So, in this problem

 .

First, given the volume, you need to find the width and length. The volume of a right, rectangular prism can be found using 

, so , where  represents the length and  represents the width.

Solving for , you get 

So, the width of the prism is 3 feet.  

Remember that the length is twice the width, so the length is 6 feet.  

Now you may use the formula for finding the diagonal: 

. So, .

First, use the volume formula,

 to find the missing length and width.  

Since  cannot be a negative value is it represents a length of a prism, we know .  So the length is 2 meters, and therefore the width is 5 meters.  

Now you can plug in the length, width and height into the formula for finding the diagonal of the prism.

 .

First, use the volume to find the missing height and length.  

Since the length is three times the height, use  to represent the length and  to represent the height.  

So, .  

So the height of the prism is 2 centimeters, and the length is 6 centimeters.

Use these values to now solve for the diagonal distance.  

.

Use the surface area, 280 square inches, and the formula for finding the surface area of a right, rectangular prsim  to find the missing length and width measurements.  

So, 

Since  represents the length of a solid figure, we must assume , rather than the negative value.  

So, the width of the figure is 2 inches and the length is 10 inches.

Now, use the formula for finding the diagonal of a right, rectangular prism:

.

Use the surface area of the prism to find the missing length, width and height.

So, the prism's length is 1 meter, the width is 2 meters and the height is 4 meters.  

Now you can find the diagonal distance using those values.  

.

Examine the diagram below.

 is a right triangle with legs of length , so it is an isosceles - or 45-45-90 - right triangle. By the 45-45-90 Triangle Theorem its hypotenuse measures

 is a right triangle with legs of lengths  and , so the length of its hypotenuse is

.

, the diagonal in question, has length , not .