If the length of each side of a regular tetrahedron is 12 cm, then the volume of the tetrahedron is

Options

A. `144sqrt(2)` cu. Cm
B. `72sqrt(2)` cu. Cm
C. `8sqrt(2)` cu. Cm
D. `12sqrt(2)` cu. Cm

Answer

To find the volume of a regular tetrahedron with each side measuring 12 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 side of the tetrahedron.

### Step 1: Identify the side length
Given that the length of each side \( A = 12 \) cm.

### Step 2: Substitute the value into the formula
Now, substitute \( A = 12 \) cm into the volume formula:

\[ V = \frac{12^3}{6\sqrt{2}} \]

### Step 3: Calculate \( 12^3 \)
Calculate \( 12^3 \):

\[ 12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728 \]

### Step 4: Substitute \( 12^3 \) back into the formula
Now substitute \( 1728 \) back into the volume formula:

\[ V = \frac{1728}{6\sqrt{2}} \]

### Step 5: Simplify the fraction
First, simplify \( \frac{1728}{6} \):

\[ \frac{1728}{6} = 288 \]

So now we have:

\[ V = \frac{288}{\sqrt{2}} \]

### Step 6: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{2} \):

\[ V = \frac{288\sqrt{2}}{2} = 144\sqrt{2} \]

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

\[ V = 144\sqrt{2} \text{ cm}^3 \]

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