The total surface area of a regular triangular pyramid with each edge of length 1 cm is

Options

A. `4sqrt(3) cm^(2)`
B. `(4)/(3)sqrt(3) cm^(2)`
C. `sqrt(3) cm^(2)`
D. `4 cm^(2)`

Answer

To find the total surface area of a regular triangular pyramid (also known as a tetrahedron) with each edge of length 1 cm, follow these steps:

### Step 1: Understand the Structure
A regular triangular pyramid consists of a triangular base and three triangular faces. All faces are equilateral triangles.

### Step 2: Calculate the Area of One Equilateral Triangle
The formula for the area \( A \) of an equilateral triangle with side length \( s \) is given by:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
In our case, each side length \( s = 1 \) cm.

### Step 3: Substitute the Side Length into the Area Formula
Substituting \( s = 1 \) cm into the area formula:
\[ A = \frac{\sqrt{3}}{4} (1)^2 = \frac{\sqrt{3}}{4} \text{ cm}^2 \]

### Step 4: Calculate the Total Surface Area
Since the pyramid has 4 equilateral triangles (1 base triangle and 3 lateral triangles), the total surface area \( T \) is:
\[ T = 4 \times A = 4 \times \frac{\sqrt{3}}{4} = \sqrt{3} \text{ cm}^2 \]

### Step 5: Conclusion
Thus, the total surface area of the regular triangular pyramid is:
\[ \text{Total Surface Area} = \sqrt{3} \text{ cm}^2 \]

### Final Answer
The total surface area of the regular triangular pyramid with each edge of length 1 cm is \( \sqrt{3} \) cm².

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