Cube Geometry Problem: Comparing Face Diagonal to Edge Length

To solve this problem, we will follow these steps:

• Step 1: Establish the geometric relationship between the edge and diagonal.
• Step 2: Use algebra to explore if equality is possible.

Now, let's proceed through each step:

Step 1: For a cube with edge length $a$, the face is a square with sides $a$, and its diagonal $d$ can be found using the Pythagorean theorem:

$d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}$

Step 2: We consider if it is possible for the edge length to equal the diagonal. Thus, we set

$a = a\sqrt{2}$

Dividing both sides by $a$, assuming $a \neq 0$, we get:

$1 = \sqrt{2}$

However, this is not correct since $\sqrt{2} \approx 1.414$ is not equal to $1$. This shows that it is impossible for the edge length and the diagonal of the face to be equal.

Therefore, the solution to the problem is No.