We solve for the side of the cube by deriving it from the volume of a cube formula:

\(\displaystyle V=s^3\)

\(\displaystyle s=\sqrt[3]{V}\)

\(\displaystyle s=\sqrt[3]{27}\)

\(\displaystyle s=3\)

To find the surface area of a cube, use the following formula:

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

Where s is our side length. 

Rearrange for s to get the following:

\(\displaystyle s=\sqrt{\frac{SA}{6}}\)

Plug in 54 for SA and solve

\(\displaystyle s=\sqrt{\frac{54}{6}}=\sqrt{9}=3\)

Don't forget your units, in this case centimeters so we get 3 centimeters

To solve for the side length of the cube, we need to use the formula for surface area. There are six faces on a cube, so its total surface area is just six times the area of one of its square faces, which is given by the side length squared:

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

\(\displaystyle 216=6L^2\)

\(\displaystyle L^2=\frac{216}{6}=36\)

\(\displaystyle L=\sqrt{36}=6\)

Volume of a cube is equal to the cube of its side. 343 is equal to 7 cubed, so the length of the whole cube is 7 cm. The cube is made up of 27 smaller cubes though. That means that one face of the cube is made up of 9 cubes and one edge of the whole cube is made up of 3 small cubes. That means that the length of one small cube is equal to the total length of one side divided by the numbers of cubes per side

\(\displaystyle \small V=s^3\)

\(\displaystyle \small \sqrt[3]{343}=7\)

Then we get our final answer by doing:

\(\displaystyle \small \frac{7}{3}cm\)

We are given the length of the diagonal of the cube.

Therefore we can find the length of an edge by using the formula

 \(\displaystyle d=e\sqrt{3}\) or \(\displaystyle e=\frac{d}{\sqrt{3}}\), and \(\displaystyle \frac{3}{\sqrt{3}}=\frac{3}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=\sqrt{3}\).

Therefore, the final answer is \(\displaystyle \sqrt{3}\).

To find side length, you must use the equation for volume of a cube and solve for \(\displaystyle s\).

\(\displaystyle V=s^3\Rightarrow s=\sqrt[3]{V}\)

Thus,

\(\displaystyle s=\sqrt[3]{64}=4\)

The diagonal of a cube extends from one of its corners diagonally through the cube to the opposite corner, so it can be thought of as the hypotenuse of a right triangle formed by the height of the cube and the diagonal of its base. First we must find the diagonal of the base, which will be the same as the diagonal of any face of the cube, by applying the Pythagorean theorem:

\(\displaystyle a^2+b^2=c^2\)

\(\displaystyle 3^2+3^2=c^2\)

\(\displaystyle c^2=18\rightarrow c=\sqrt{18}=3\sqrt{2}\)

Now that we know the length of the diagonal of any face on the cube, we can use the Pythagorean theorem again with this length and the height of the cube, whose hypotenuse is the length of the diagonal for the cube:

\(\displaystyle c^2+l^2=d^2\)

\(\displaystyle (3\sqrt{2})^2+3^2=d^2\)

\(\displaystyle d^2=27\rightarrow d=\sqrt{27}=3\sqrt{3}\)

To find the diagonal of a cube we can apply the formula \(\displaystyle d=e\sqrt{3}\), where \(\displaystyle d\) is the length of the diagonal and where \(\displaystyle e\) is the length of an edge of the cube.

Since we are given an area of a face of the cube, we can find the length of an edge simply by taking its square root.

\(\displaystyle 9=s^2 \rightarrow \sqrt 9 =s \rightarrow 3=s\)

Here the length of an edge is 3.

Thefore the final andwer is \(\displaystyle d=e\sqrt3=3\sqrt{3}\).

To find the length of the diagonal of the cube, we can apply the formula, however, we firstly need to find the length of an edge, by applying the formula for the diagonal of the square.

 \(\displaystyle d=s\sqrt{2}\) where \(\displaystyle d\) is the diagonal of face ABCD, and \(\displaystyle s\), the length of one of the side of this square.

The length of \(\displaystyle s\) must be \(\displaystyle \sqrt{2}\), which is the length of the edges of the square.

Therefore we can now use the formula for the length of the diagonal of the cube: 

\(\displaystyle e\sqrt{3}\), where \(\displaystyle e\) is the length of an edge.

Since \(\displaystyle e=\sqrt{2}\), we get the final answer \(\displaystyle \sqrt{6}\).

The diagonal \(\displaystyle d\) of a cube with an edge length \(\displaystyle s\) can be defined by the equation \(\displaystyle d=s\sqrt{3}\). Given \(\displaystyle s=5\) in this instance, \(\displaystyle d=5\sqrt{3}\).