Let OABC be a regular tetrahedron with side length unity, then its volume (in cubic units) is

Options

A. the length of perpendicular from one vertex to opposite face is `sqrt(2/3)`
B. the perpendicular distance from mid-point `overline(OA)` to the plane ABC is `(1)/(sqrt(6))`
C. the angle between two skew edges to `(phi)/(2)`
D. the distance of centroid of the tetrahedron form any vertex is `sqrt(3/8)`.

Answer

To find the volume of a regular tetrahedron with a side length of 1 unit, we can use the formula for the volume of a tetrahedron:

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

where \(a\) is the length of a side of the tetrahedron.

### Step-by-Step Solution:

1. **Identify the side length**:
Given that the side length \(a\) of the tetrahedron is 1 unit.

2. **Substitute the side length into the volume formula**:
We substitute \(a = 1\) into the volume formula:
\[ V = \frac{1^3}{6\sqrt{2}} \]

3. **Calculate \(1^3\)**:
\[ 1^3 = 1 \]

4. **Substitute back into the formula**:
Now we have:
\[ V = \frac{1}{6\sqrt{2}} \]

5. **Simplify the expression**:
The volume of the tetrahedron is:
\[ V = \frac{1}{6\sqrt{2}} \text{ cubic units} \]

Thus, the volume of the regular tetrahedron OABC with side length 1 unit is \(\frac{1}{6\sqrt{2}}\) cubic units.