To find the length of the diagonal, use the formula for a triangle:

The length of the diagonal is .

First, you need to find the side length of this cube. We know that the volume is:

, where is the side length.

Therefore, based on our data, we can say:

Solving for by taking the cube-root of both sides, we get:

Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

and

Solve this by using the distance formula. This is very easy since one point is all s. It is merely:

This is approximately .

The diagonal of a cube is simply given by:

Where is the side length of the cube.

So since our

First, you need to find the side length of this cube. We know that the surface area is defined by:

, where is the side length. (This is because the cube is sides of equal area).

Therefore, based on our data, we can say:

Take the square root of both sides and get:

Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

and

Solve this by using the distance formula. This is very easy since one point is all s. It is merely:

This is approximately .

Find the diagonal of one face of the cube using the Pythagorean Theorem applied to a triangle formed by two sides of that face ( and ) and the diagonal itself ():

This diagonal is now the base of a new right triangle (call this ). The height of that triangle is an edge of the cube that runs perpendicular to this diagonal (call this ). The third side of the triangle formed by and is a line from one corner of the cube to the other, i.e., the cube's diagonal (call this ). Use the Pythagorean Theorem again with the triangle formed by , , and to find the length of this diagonal.

Write the equation for the volume of a cube. Substitute the volume to find the side length, .

Write the equation for finding diagonals given an edge length for a cube.

Substitute the side length to find the diagonal length.

The answer is .

There is a formula for the length of a cube's diagonal given the side length. However, we might not remember that formula as it is less common. However, we can also find the length using the Pythagorean Theorem.

But first, we need to find the side length. We know the volume is 64. Our formula for volume is

Substituting gives

Taking the cube root gives us a side length of 4. Now let's look at our cube.

We need to begin by finding the length of the diagonal of the bottom face of our cube (the green segment). This can be done either by using the Pythagorean Theorem or by realizing that the right triangle is in fact a 45-45-90 triangle. Either way, we realize that our diagonal (the hypotenuse) is .

We now seek to find the diagonal of the cube (the blue segment). We do this by looking at the right triangle formed by it, the left vertical edge, and the face diagonal we just found. This time our only recourse is to do the Pythagorean Theorem.

In general, the formula for the diagonal of a cube with side length is

The length of our diagonal is

Write the volume of a cube and substitute the given volume to find a side length.

Write the diagonal formula for a cube and substitute the side length.

Write the equation for finding the volume of a cube, and substitute in the volume.

Write the diagonal equation for cubes and substitute in the given length.

Write the surface area formula for cubes and substitute the given area.

Write the diagonal formula for cubes and substitute the side length.