The area of the base of a right prism whose base is an equilateral triangle is `9sqrt(3) cm^(2)` . If the height of the prism is 12 cm , then what is its lateral surface area ?

Options

A. `212 cm^(2)`
B. `21 cm^(2)`
C. `216 cm^(2)`
D. `222 cm^(2)`

Answer

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

### Step 1: Understand the formula for lateral surface area
The lateral surface area (LSA) of a prism is given by the formula:
\[ \text{LSA} = \text{Perimeter of the base} \times \text{Height} \]

### Step 2: Find the perimeter of the base
Since the base of the prism is an equilateral triangle, we need to find the length of one side of the triangle. We know the area of the equilateral triangle is given by:
\[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]
where \( a \) is the length of a side of the triangle.

Given the area of the base is \( 9\sqrt{3} \, \text{cm}^2 \), we can set up the equation:
\[ 9\sqrt{3} = \frac{\sqrt{3}}{4} a^2 \]

### Step 3: Solve for \( a^2 \)
To eliminate \( \sqrt{3} \) from both sides, we divide both sides by \( \sqrt{3} \):
\[ 9 = \frac{1}{4} a^2 \]

Now, multiply both sides by 4:
\[ 36 = a^2 \]

### Step 4: Find \( a \)
Now, take the square root of both sides to find \( a \):
\[ a = \sqrt{36} = 6 \, \text{cm} \]

### Step 5: Calculate the perimeter of the base
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]
Substituting the value of \( a \):
\[ P = 3 \times 6 = 18 \, \text{cm} \]

### Step 6: Calculate the lateral surface area
Now, we can use the height of the prism, which is given as 12 cm, to find the lateral surface area:
\[ \text{LSA} = P \times \text{Height} \]
\[ \text{LSA} = 18 \times 12 = 216 \, \text{cm}^2 \]

### Final Answer
The lateral surface area of the prism is \( 216 \, \text{cm}^2 \).

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