The base of a right pyramid is an equilateral triangle, each side of which is `6sqrt(3)` cm long and its height is 4 cm. Find the total surface area of the pyramid in `cm^(2)`.

Answer

To find the total surface area of the right pyramid with an equilateral triangle base, we will follow these steps:

### Step 1: Calculate the area of the base
The base of the pyramid is an equilateral triangle with each side measuring \(6\sqrt{3}\) cm. The formula for the area \(A\) of an equilateral triangle with side length \(a\) is given by:

\[ A = \frac{\sqrt{3}}{4} a^2 \]

Substituting \(a = 6\sqrt{3}\):

\[ A = \frac{\sqrt{3}}{4} (6\sqrt{3})^2 \]
\[ = \frac{\sqrt{3}}{4} \cdot 36 \cdot 3 \]
\[ = \frac{\sqrt{3}}{4} \cdot 108 \]
\[ = 27\sqrt{3} \text{ cm}^2 \]

### Step 2: Calculate the slant height of the pyramid
To find the lateral surface area, we first need the slant height of the pyramid. The height of the pyramid is given as \(4\) cm. We can find the slant height \(l\) using the Pythagorean theorem in the triangle formed by the height, half the base, and the slant height.

The length of the median of the equilateral triangle (which is also the height of the triangle) can be calculated as:

\[ \text{Median} = \frac{\sqrt{3}}{2} a = \frac{\sqrt{3}}{2} \cdot 6\sqrt{3} = 9 \text{ cm} \]

The half base length is:

\[ \text{Half base} = \frac{6\sqrt{3}}{2} = 3\sqrt{3} \text{ cm} \]

Now applying the Pythagorean theorem:

\[ l = \sqrt{(\text{Height})^2 + (\text{Half base})^2} \]
\[ l = \sqrt{4^2 + (3\sqrt{3})^2} \]
\[ = \sqrt{16 + 27} = \sqrt{43} \text{ cm} \]

### Step 3: Calculate the lateral surface area
The lateral surface area (LSA) of the pyramid can be calculated using the formula:

\[ \text{LSA} = \frac{1}{2} \times \text{Perimeter of base} \times \text{Slant height} \]

The perimeter \(P\) of the base (equilateral triangle) is:

\[ P = 3 \times (6\sqrt{3}) = 18\sqrt{3} \text{ cm} \]

Now substituting the values into the LSA formula:

\[ \text{LSA} = \frac{1}{2} \times (18\sqrt{3}) \times \sqrt{43} \]
\[ = 9\sqrt{3} \times \sqrt{43} \text{ cm}^2 \]

### Step 4: Calculate the total surface area
The total surface area (TSA) of the pyramid is the sum of the area of the base and the lateral surface area:

\[ \text{TSA} = \text{Area of base} + \text{LSA} \]
\[ = 27\sqrt{3} + 9\sqrt{3} \times \sqrt{43} \]

### Final Result
Thus, the total surface area of the pyramid is:

\[ \text{TSA} = 27\sqrt{3} + 9\sqrt{3}\sqrt{43} \text{ cm}^2 \]