Question

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

Answer 1

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:

\[
\boxed{6} \text{ cm}^3
\]