If the altitude of a right prism is 10 cm and its base is an equilateral triangle of side 12 cm, then its total surface area (in `cm^(2)`) is

Options

A. `(5 + 3sqrt(3))`
B. `36sqrt(3)`
C. 360
D. `72(5+sqrt(3))`

Answer

To find the total surface area of a right prism with an equilateral triangle as its base, we can follow these steps:

### Step 1: Calculate the area of the base (equilateral triangle)
The formula for the area \( A \) of an equilateral triangle with side length \( a \) is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Here, the side length \( a = 12 \) cm. Thus, substituting the value:
\[ A = \frac{\sqrt{3}}{4} \times (12)^2 \]
\[ A = \frac{\sqrt{3}}{4} \times 144 \]
\[ A = 36\sqrt{3} \text{ cm}^2 \]

### Step 2: Calculate the perimeter of the base
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]
Substituting \( a = 12 \) cm:
\[ P = 3 \times 12 = 36 \text{ cm} \]

### Step 3: Calculate the total surface area of the prism
The total surface area \( TSA \) of a prism is given by the formula:
\[ TSA = 2 \times \text{Base Area} + \text{Perimeter} \times \text{Height} \]
Here, the height \( h = 10 \) cm. Substituting the values we calculated:
\[ TSA = 2 \times (36\sqrt{3}) + 36 \times 10 \]
\[ TSA = 72\sqrt{3} + 360 \]

### Step 4: Final expression for the total surface area
Thus, the total surface area of the prism is:
\[ TSA = 72\sqrt{3} + 360 \text{ cm}^2 \]