If the apothem of a regular hexagon is a, find each side of the hexagon.

Options

A. A)`3a/sqrt2`
B. B)`sqrt3a`
C. C)0.67a
D. D)`2/sqrt3a`

Answer

To find the length of each side of a regular hexagon given its apothem \( a \), we can follow these steps:

### Step 1: Understand the relationship between the apothem and the side length
In a regular hexagon, the apothem \( a \) is related to the side length \( k \) by the formula:
\[ a = \frac{k \sqrt{3}}{2} \]
This formula arises from the geometry of the hexagon, where the apothem is the height of the equilateral triangles that make up the hexagon.

### Step 2: Rearrange the formula to solve for \( k \)
To find the side length \( k \), we can rearrange the formula:
\[ k = \frac{2a}{\sqrt{3}} \]

### Step 3: Write down the final expression for the side length
Thus, the length of each side of the hexagon is given by:
\[ k = \frac{2a}{\sqrt{3}} \]

### Step 4: Identify the correct option
Now, we need to compare this result with the options given in the question. The expression \( \frac{2a}{\sqrt{3}} \) matches with option (d).

### Final Answer:
The length of each side of the hexagon is:
\[ \frac{2a}{\sqrt{3}} \]