The formula for the volume is given by

\(\displaystyle (side)^3=volume\)

Since we have the volume, we must take the cube root of the volume to find the length of any one side (since it is a cube, all of the sides are equal).

\(\displaystyle \sqrt[3]{(side)^3}=side=\sqrt[3]{volume}\)

Plugging in 216 for the volume, we end up with

\(\displaystyle side=6\ cm\)

Because this is a cube, it's helpful to remember that the value of the diagonal of one face is the same length for the other five faces. Additionally, the length of one edge will be the same length of all the cube's other edges. This helps relieve the stress of there being more than one possible right answer. 

To find the edge, we're looking for the length of one of the sides of the square's faces. The problem can be seen in a simplified square convention: 

The diagonal merely splits a square into two right 45-45-90 triangles. Finding the length of one the sides of the squares can be solved either through trigonometric functions or the rules for 45-45-90 triangles. 

Using the rules for 45-45-90 triangles:

The hypotenuse of the created triangle is \(\displaystyle 9\sqrt2\), which can be set equal to \(\displaystyle s\sqrt2\) to solve for \(\displaystyle s\), which in this case will give us the length of one of the cube's edges. 

\(\displaystyle 9\sqrt{2}=s\sqrt{2}\)

\(\displaystyle \frac{9\sqrt{2}}{\sqrt{2}} = \frac{s\sqrt2}\sqrt2{}\)

\(\displaystyle {\color{Blue} s= 9}\)

Therefore, the edge length of the cube is \(\displaystyle 9\:units\).

The surface area of a cube can be represented as \(\displaystyle 6s^2\), since a cube has six sides and the surface area of each side is represented by its length multiplied by its width, which for a cube is \(\displaystyle s^2\), since all of its edges are the same length.

We can substitute \(\displaystyle 24\:cm^2\) into this equation and then solve for \(\displaystyle a\):

\(\displaystyle SA=6s^2\)

\(\displaystyle 24=6a^2\)

\(\displaystyle 4=a^2\)

\(\displaystyle a=2\)

So, one edge of this cube is \(\displaystyle 2\:cm\) in length.

Since the volume of a cube is length times width times height, with every measurement being the same, we just need to take the cube root of the volume:

 \(\displaystyle ^3\sqrt{43}=3.50\).

Rounded to the nearest tenth, the length is 3.5 feet.

The only information that is given is that the diagonal of one of the faces of the cube is \(\displaystyle 4\sqrt5\:units\). Because this is a cube, this is true for the rest of the five faces. All the edges will also be the same length, meaning this eliminates the possibility of more than one correct answer.

The edge of the cube can be solved for using the Pythagorean Theorem because the diagonal creates two right triangles. Or, if you're comfortable with it, you can remember that the diagonal creates two \(\displaystyle 45-45-90\) special right triangles that have their own rules with respect to solving for sides.

Using the Pythagorean Theorem, \(\displaystyle a^2+b^2=c^2\), we can simplify the equation according to information we have and can deduce the correct answer.

The variables \(\displaystyle a\) and \(\displaystyle b\) refer to the legs of the right triangles. Because this is a cube, we can deduce that the length of the legs will be the same. Therefore, \(\displaystyle a=b=x\).  That means the Pythagorean Theorem (for this case) can be rewritten as \(\displaystyle x^2+x^2=c^2\)

Looking back at the problem, the only information given is the hypotenuse for one of the two triangles. This value can be substituted in for \(\displaystyle c\). Then, as we solve for \(\displaystyle x\), we will obtain the answer for the question: the length of the edge.

\(\displaystyle x^2+x^2=(4\sqrt{5})^2\)

\(\displaystyle 2x^2=80\)

\(\displaystyle x^2=\frac{80}{2}\)

\(\displaystyle x^2=40\)

\(\displaystyle \sqrt{x^2}=\sqrt{40}\)

\(\displaystyle x=\sqrt{40}\)

\(\displaystyle x=\sqrt{10\cdot2\cdot2}\)

\(\displaystyle x=2\sqrt{10}\)

Therefore, the edge of the cube is \(\displaystyle 2\sqrt{10}\:units\). 

Since cubes have side lengths that are equal and we find the volume by \(\displaystyle V=s*s*s\), then the side length of a cube with a volume of \(\displaystyle 512\hspace{1mm}m^3\) is simply \(\displaystyle 512\hspace{1mmm^3}=s*s*s\). In other words what number multiplied three times gives 512? Take the cube root of 512 to get \(\displaystyle \mathbf{8\hspace{1mm} meters}\).

Recall how to find the surface area of a cube:

\(\displaystyle \text{Surface Area}=6(\text{side length})^2\)

Since the question asks you to find the length of a side of this cube, rearrange the equation.

\(\displaystyle \text{side length}^2=\frac{\text{Surface Area}}{6}\)

\(\displaystyle \text{side length}=\sqrt{\frac{\text{Surface Area}}{6}}\)

Substitute in the given surface area to find the side length.

\(\displaystyle \text{side length}=\sqrt{\frac{162}{6}}\)

Simplify.

\(\displaystyle \text{side length}=\sqrt{27}\)

Reduce.

\(\displaystyle \text{side length}=3\sqrt3\)

Recall how to find the surface area of a cube:

\(\displaystyle \text{Surface Area}=6(\text{side length})^2\)

Since the question asks you to find the length of a side of this cube, rearrange the equation.

\(\displaystyle \text{side length}^2=\frac{\text{Surface Area}}{6}\)

\(\displaystyle \text{side length}=\sqrt{\frac{\text{Surface Area}}{6}}\)

Substitute in the given surface area to find the side length.

\(\displaystyle \text{side length}=\sqrt{\frac{96}{6}}\)

Simplify.

\(\displaystyle \text{side length}=\sqrt{16}\)

Reduce.

\(\displaystyle \text{side length}=4\)

Recall how to find the surface area of a cube:

\(\displaystyle \text{Surface Area}=6(\text{side length})^2\)

Since the question asks you to find the length of a side of this cube, rearrange the equation.

\(\displaystyle \text{side length}^2=\frac{\text{Surface Area}}{6}\)

\(\displaystyle \text{side length}=\sqrt{\frac{\text{Surface Area}}{6}}\)

Substitute in the given surface area to find the side length.

\(\displaystyle \text{side length}=\sqrt{\frac{864}{6}}\)

Simplify.

\(\displaystyle \text{side length}=\sqrt{144}\)

Reduce.

\(\displaystyle \text{side length}=12\)

Recall how to find the surface area of a cube:

\(\displaystyle \text{Surface Area}=6(\text{side length})^2\)

Since the question asks you to find the length of a side of this cube, rearrange the equation.

\(\displaystyle \text{side length}^2=\frac{\text{Surface Area}}{6}\)

\(\displaystyle \text{side length}=\sqrt{\frac{\text{Surface Area}}{6}}\)

Substitute in the given surface area to find the side length.

\(\displaystyle \text{side length}=\sqrt{\frac{750}{6}}\)

Simplify.

\(\displaystyle \text{side length}=\sqrt{125}\)

Reduce.

\(\displaystyle \text{side length}=5\sqrt5\)