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,

$$A\approx 3\times 3.732\times 0.2678\approx 3\times 1.000\approx 3(\text{approximately 3 when calculated correctly})$$

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:

$$\boxed{{\displaystyle 6}}{\text{ cm}}^3$$