Length of each edge of a regular tetrahedron is 1 cm. Its volume is :

Options

A. `(sqrt(3))/(12)` cu. cm.
B. `(1)/(3)sqrt(3)` cu. cm
C. `(sqrt(2))/(6)` cu. cm.
D. `(1)/(12)sqrt(2)` cu. cm

Answer

To find the volume of a regular tetrahedron with each edge measuring 1 cm, we can use the formula for the volume of a tetrahedron:

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

where \( a \) is the length of each edge.

### Step 1: Identify the edge length
Given that the length of each edge \( a = 1 \) cm.

### Step 2: Substitute the edge length into the volume formula
Substituting \( a = 1 \) cm into the volume formula:

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

### Step 3: Simplify the expression
Calculating \( 1^3 \):

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

### Step 4: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by \( \sqrt{2} \):

\[ V = \frac{1 \cdot \sqrt{2}}{6\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{6 \cdot 2} = \frac{\sqrt{2}}{12} \]

### Step 5: Write the final answer
Thus, the volume of the regular tetrahedron is:

\[ V = \frac{\sqrt{2}}{12} \text{ cm}^3 \]

### Summary of the solution:
The volume of the regular tetrahedron with edge length 1 cm is \( \frac{\sqrt{2}}{12} \text{ cm}^3 \).

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