Each edge of a regular tetrahedron is 4 cm. its volume (in cubic cm) is

Options

A. `(16sqrt(3))/(3)`
B. `16sqrt(3)`
C. `(16sqrt(2))/(3)`
D. `16sqrt(2)`

Answer

To find the volume of a regular tetrahedron with each edge measuring 4 cm, we will use the formula for the volume of a regular tetrahedron:

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

where \(s\) is the length of an edge.

### Step 1: Identify the edge length
The edge length \(s\) is given as 4 cm.

### Step 2: Calculate the cube of the edge length
We need to calculate \(s^3\):

\[ s^3 = 4^3 = 4 \times 4 \times 4 = 64 \text{ cm}^3 \]

### Step 3: Substitute into the volume formula
Now we substitute \(s^3\) into the volume formula:

\[ V = \frac{64}{6\sqrt{2}} \]

### Step 4: Simplify the volume expression
To simplify \(\frac{64}{6\sqrt{2}}\), we can multiply the numerator and the denominator by \(\sqrt{2}\):

\[ V = \frac{64 \cdot \sqrt{2}}{6 \cdot 2} = \frac{64\sqrt{2}}{12} \]

### Step 5: Further simplify the fraction
Now we can simplify \(\frac{64\sqrt{2}}{12}\):

\[ V = \frac{16\sqrt{2}}{3} \text{ cm}^3 \]

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

\[ V = \frac{16\sqrt{2}}{3} \text{ cm}^3 \]

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