In a regular tetrahedron, if the distance between the mid points of opposite edges is unity, its volume is

Options

A. (a)`(1)/(3)`
B. (b)`(1)/(2)`
C. (c)`(1)/(sqrt(2))`
D. (d)`(1)/(6sqrt(2))`

Answer

To find the volume of a regular tetrahedron given that the distance between the midpoints of opposite edges is unity, we can follow these steps:

### Step 1: Understand the properties of a regular tetrahedron
A regular tetrahedron has all its edges of equal length. Let the length of each edge be denoted as \( A \).

### Step 2: Find the distance between midpoints of opposite edges
In a regular tetrahedron, the distance between the midpoints of opposite edges can be derived from the geometry of the tetrahedron. The formula for the distance \( d \) between the midpoints of two opposite edges is given by:

\[ d = \frac{A}{\sqrt{2}} \]

### Step 3: Set the distance equal to unity
According to the problem, this distance is given as unity (1). Therefore, we set up the equation:

\[ \frac{A}{\sqrt{2}} = 1 \]

### Step 4: Solve for \( A \)
To find \( A \), we can rearrange the equation:

\[ A = \sqrt{2} \]

### Step 5: Calculate the volume of the tetrahedron
The volume \( V \) of a regular tetrahedron with edge length \( A \) is given by the formula:

\[ V = \frac{A^3}{6\sqrt{2}} \]

### Step 6: Substitute \( A \) into the volume formula
Now, substituting \( A = \sqrt{2} \) into the volume formula:

\[ V = \frac{(\sqrt{2})^3}{6\sqrt{2}} = \frac{2\sqrt{2}}{6\sqrt{2}} = \frac{2}{6} = \frac{1}{3} \]

### Final Answer
Thus, the volume of the tetrahedron is:

\[ \frac{1}{3} \]