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. 

You own a Rubik's cube with a volume of . What is the edge length of the cube?

To solve for edge length, think of the volume of a cube formula:

Now, we have the volume, so just rearrange it to solve for side length:

General formula for the diagonal of a cube if each side of the cube = s

Use Pythagorean Theorem to get the diagonal across the base:

s² + s² = h²

And again use Pythagorean Theorem to get cube’s diagonal, then solve for d:

h² + s² = d²

s² + s² + s² = d²

3 * s² = d²

d = √ (3 * s²) = s √3

So, if s = 3 then the answer is 3√3

Since the diagonal of the cube is a line segment that goes through the center of the cube (and also the circumscribed sphere), it is clear that the diagonal of the cube is also the diameter of the sphere.  Since the radius = 1, the diameter = 2.

The first thing necessary is to determine the dimensions of the cube.  This can be done using the volume formula for cubes: V = s³, where s is the length of the cube. For our data, this is:

s³ = 512, or (taking the cube root of both sides), s = 8.

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (8,8,8).  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) = √( (8)² + (8)²  + (8)²) = √( 64 + 64 + 64) = √(64 * 3) = 8√(3)

The first thing necessary is to determine the dimensions of the cube.  This can be done using the volume formula for cubes: V = s³, where s is the length of the cube. For our data, this is:

s³ = 1728, or (taking the cube root of both sides), s = 12.

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (12,12,12).  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) = √( (12)² + (12)²  + (12)²) = √( 144 + 144 + 144) = √(3 * 144) = 12√(3) = 12√(3)

 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 .

This problem is relatively simple. We know that the volume of a cube is equal to s³, where s is the length of a given side of the cube. Therefore, to find our dimensions, we merely have to solve s³ = 1728. Taking the cubed root, we get s = 12.

Since the sides of a cube are all the same, the surface area of the cube is equal to 6 times the area of one face. For our dimensions, one face has an area of 12 * 12 or 144 in². Therefore, the total surface area is 6 * 144 = 864 in².

If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 9 ft high, we know 18 x 15 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 18 x 9 and 15 x 9. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:

2 * (18 * 9 + 15 * 9) = 2 * (162 + 135) = 2 * 297 = 594 ft²

Now, we merely need to calculate the area "taken out" of the walls:

For the door: 3 * 7 = 21 ft² 

For the windows: 2 * (2 * 5) = 20 ft²

The total wall space is therefore: 594 – 21 – 20 = 553 ft²