A regular 12 - sided polygon is the cross section of a prism of length 2 cm. Calculate the volume of the prism.

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 ]