The length of each edge of a regular tetrahedron is 12 cm. The area (in sq. cm) of the total surface of the tetrahedron is

Options

A. `288sqrt(3)`
B. `144sqrt(2)`
C. `108 sqrt(3)`
D. `144sqrt(3)`

Answer

To find the total surface area of a regular tetrahedron with each edge measuring 12 cm, we can follow these steps:

### Step 1: Understand the formula for the total surface area of a tetrahedron.
The total surface area (TSA) of a regular tetrahedron can be calculated using the formula:
\[ \text{TSA} = \sqrt{3} \times a^2 \]
where \( a \) is the length of each edge of the tetrahedron.

### Step 2: Substitute the edge length into the formula.
Given that the length of each edge \( a = 12 \) cm, we substitute this value into the formula:
\[ \text{TSA} = \sqrt{3} \times (12)^2 \]

### Step 3: Calculate \( (12)^2 \).
First, we calculate \( (12)^2 \):
\[ (12)^2 = 144 \]

### Step 4: Multiply by \( \sqrt{3} \).
Now we substitute back into the formula:
\[ \text{TSA} = \sqrt{3} \times 144 \]

### Step 5: Calculate the total surface area.
To find the numerical value, we can use the approximate value of \( \sqrt{3} \approx 1.732 \):
\[ \text{TSA} \approx 1.732 \times 144 \]
Calculating this gives:
\[ \text{TSA} \approx 248.832 \text{ sq. cm} \]

### Step 6: Round the answer if necessary.
Depending on the context, you might round this to two decimal places or leave it as is. For this case, we can round it to:
\[ \text{TSA} \approx 248.83 \text{ sq. cm} \]

### Final Answer:
The total surface area of the tetrahedron is approximately \( 248.83 \) sq. cm.

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