The base of a prism is a regular hexagon. If every edge of the prism measures 1 m, then the volume of the prism is :

Options

A. `(3sqrt(2))/(2) m^(3) `
B. `(3sqrt(3))/(2) m^(3) `
C. `(6 sqrt(2))/(5) m^(3) `
D. `(5 sqrt(3))/(2) m^(3)`

Answer

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

### Step 1: Calculate the area of the hexagonal base

A regular hexagon can be divided into 6 equilateral triangles. The formula for the area \( A \) of a regular hexagon with side length \( a \) is given by:

\[ A = \frac{3\sqrt{3}}{2} a^2 \]

In this case, each edge of the prism measures 1 m, so \( a = 1 \) m.

Substituting the value of \( a \):

\[ A = \frac{3\sqrt{3}}{2} (1)^2 = \frac{3\sqrt{3}}{2} \text{ m}^2 \]

### Step 2: Determine the height of the prism

Since the prism is a right prism and every edge measures 1 m, the height \( h \) of the prism is also 1 m.

### Step 3: Calculate the volume of the prism

The volume \( V \) of a prism is calculated using the formula:

\[ V = \text{Base Area} \times \text{Height} \]

Substituting the values we found:

\[ V = \left(\frac{3\sqrt{3}}{2}\right) \times 1 = \frac{3\sqrt{3}}{2} \text{ m}^3 \]

Thus, the volume of the prism is:

\[ \boxed{\frac{3\sqrt{3}}{2} \text{ m}^3} \]

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