A prism has a regular hexagonal base with side 6 cm. If the total surface area of prism is `216 sqrt""3 cm^(2)` . then what is the height (in cm) of prism?

Options

A. `3 sqrt""3`
B. `6 sqrt""3`
C. 6
D. 3

Answer

To find the height of the prism with a regular hexagonal base, we will follow these steps:

### Step 1: Calculate the Area of the Hexagonal Base
The formula for the area \( A \) of a regular hexagon with side length \( a \) is given by:
\[ A = \frac{3\sqrt{3}}{2} a^2 \]
Given that the side length \( a = 6 \) cm, we can substitute this value into the formula:
\[ A = \frac{3\sqrt{3}}{2} \times (6)^2 = \frac{3\sqrt{3}}{2} \times 36 = 54\sqrt{3} \text{ cm}^2 \]

### Step 2: Write the Formula for Total Surface Area of the Prism
The total surface area \( TSA \) of a prism is given by:
\[ TSA = 2 \times \text{Area of Base} + \text{Perimeter of Base} \times \text{Height} \]
For a hexagonal base, the perimeter \( P \) is:
\[ P = 6 \times a = 6 \times 6 = 36 \text{ cm} \]
Thus, the total surface area can be expressed as:
\[ TSA = 2 \times 54\sqrt{3} + 36h \]

### Step 3: Set Up the Equation with the Given Total Surface Area
We know from the problem that the total surface area is \( 216\sqrt{3} \) cm². So we can set up the equation:
\[ 216\sqrt{3} = 2 \times 54\sqrt{3} + 36h \]

### Step 4: Simplify and Solve for Height \( h \)
Calculating the left side:
\[ 216\sqrt{3} = 108\sqrt{3} + 36h \]
Now, isolate \( h \):
\[ 216\sqrt{3} - 108\sqrt{3} = 36h \]
\[ 108\sqrt{3} = 36h \]
Now, divide both sides by 36:
\[ h = \frac{108\sqrt{3}}{36} = 3\sqrt{3} \text{ cm} \]

### Conclusion
The height of the prism is \( 3\sqrt{3} \) cm.

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