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

Options

A. `3sqrt2`
B. `6sqrt2`
C. `(1)/(3sqrt2)`
D. `(1)/(6sqrt2)`

Answer

To find the volume of a regular tetrahedron with side length unity, we can use the formula for the volume of a tetrahedron given by:

\[ V = \frac{1}{6} \times \text{Base Area} \times \text{Height} \]

### Step 1: Calculate the Base Area
The base of the tetrahedron can be taken as triangle ABC. The area \( A \) of an equilateral triangle with side length \( a \) is given by:

\[ A = \frac{\sqrt{3}}{4} a^2 \]

For our tetrahedron, the side length \( a = 1 \):

\[ A = \frac{\sqrt{3}}{4} (1^2) = \frac{\sqrt{3}}{4} \]

### Step 2: Calculate the Height
The height \( h \) of the tetrahedron can be calculated using the formula for the height of a tetrahedron from one vertex to the opposite face. The height can be derived from the relationship of the tetrahedron's geometry. The height from vertex O to the base triangle ABC can be calculated as follows:

The height \( h \) of a regular tetrahedron with side length \( a \) is given by:

\[ h = \sqrt{\frac{2}{3}} a \]

Substituting \( a = 1 \):

\[ h = \sqrt{\frac{2}{3}} \cdot 1 = \sqrt{\frac{2}{3}} \]

### Step 3: Calculate the Volume
Now we can substitute the base area and height into the volume formula:

\[ V = \frac{1}{6} \times \text{Base Area} \times \text{Height} \]

Substituting the values we found:

\[ V = \frac{1}{6} \times \frac{\sqrt{3}}{4} \times \sqrt{\frac{2}{3}} \]

### Step 4: Simplify the Expression
Now, simplify the expression:

\[ V = \frac{1}{6} \times \frac{\sqrt{3}}{4} \times \sqrt{\frac{2}{3}} = \frac{\sqrt{3} \cdot \sqrt{2}}{6 \cdot 4 \cdot \sqrt{3}} = \frac{\sqrt{6}}{24} \]

### Step 5: Final Volume Calculation
Thus, the volume of the tetrahedron is:

\[ V = \frac{\sqrt{6}}{24} \]

However, we can also use the known formula for the volume of a regular tetrahedron directly:

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

Substituting \( a = 1 \):

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

### Conclusion
The volume of the regular tetrahedron with side length unity is:

\[ V = \frac{1}{6\sqrt{2}} \text{ cubic units} \]