What is the ratio of lengths of face edge, face diagonal and body diagonal of cube ?

Explanation

To find the ratio of the lengths of the face edge, face diagonal, and body diagonal of a cube, we can follow these steps:

### Step 1: Define the Length of the Edge of the Cube
Let the length of the edge of the cube be denoted as \( a \).

### Step 2: Calculate the Length of the Face Diagonal
The face diagonal of a cube can be calculated using the Pythagorean theorem. In a face of the cube, we have a right triangle formed by two edges of the cube. The face diagonal \( d_f \) can be calculated as:
\[
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