To find the volume of a prism, you can use the formula:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
In this case, the base is a regular 12-sided polygon (dodecagon), and the height of the prism is given as 2 cm.
Step 1: Calculate the Area of the Regular Dodecagon
The area \( A \) of a regular 12-sided polygon can be calculated using the formula:
\[ A = 3 \times \cot\left(\frac{\pi}{12}\right) \times s^2 \]
Where:
- \( s \) is the length of a side of the dodecagon,
- \( \cot(\theta) \) is the cotangent function.
However, we need to determine the side length \( s \). The standard side length for a unit circumradius \( R \) can be expressed as:
\[ s = R \times 2 \sin\left(\frac{\pi}{12}\right) \]
For a regular dodecagon, the circumradius \( R \) is also often taken as the distance from the center to a vertex. Let's assume a unit circumradius for simplicity unless a specific radius is provided. After finding \( s \), we can substitute back into the area formula.
Step 2: Calculate Side Length
Given the dodecagon's geometry:
\[ s = 2R \sin\left(\frac{\pi}{12}\right) \]
If we assume a unit circumradius (i.e. R = 1), then the side length is:
\[ s = 2 \sin\left(\frac{\pi}{12}\right) \]
Calculating \( \sin\left(\frac{\pi}{12}\right) \):
\(\sin\left(\frac{\pi}{12}\right) = \sin(15^\circ) \approx 0.2588\)
Then,
\[ s \approx 2 \times 0.2588 \approx 0.5176 \]
Step 3: Calculate the Area
Now substituting \( s \) back into the area formula:
\[ A = 3 \times \cot\left(\frac{\pi}{12}\right) \times s^2 = 3 \times \cot\left(15^\circ\right) \times (0.5176)^2 \]
Calculating \( \cot\left(15^\circ\right) \):
\(\cot(15^\circ) \approx 3.732\)
Then,
\[ s^2 \approx (0.5176)^2 \approx 0.2678 \]
Thus,
\[ A \approx 3 \times 3.732 \times 0.2678 \approx 3 \times 1.000 \approx 3 \quad (\text{approximately 3 when calculated correctly}) \]
Step 4: Calculate Volume of the Prism
Now we can plug the area and height into the volume formula:
\[ \text{Volume} = A \times \text{Height} = 3 \times 2 = 6 \text{ cm}^3 \]
Final Result:
The volume of the prism is:
\[ \boxed{6} \text{ cm}^3 \]