If a regular hexagon circumscribes a circle of radius r , then its perimeter is :

Options

A. `4 sqrt(3)r`
B. `6 sqrt(3) r`
C. `6r`
D. `12 sqrt(3)r`

Answer

To find the perimeter of a regular hexagon that circumscribes a circle of radius \( r \), we can follow these steps:

### Step 1: Understand the relationship between the radius and the side length of the hexagon.
A regular hexagon can be divided into 6 equilateral triangles. The radius \( r \) of the inscribed circle (the circle that touches all sides of the hexagon) is related to the side length \( a \) of the hexagon. The radius \( r \) is given by the formula:
\[ r = \frac{\sqrt{3}}{2} a \]

### Step 2: Solve for the side length \( a \).
To find the side length \( a \) in terms of the radius \( r \), we rearrange the formula:
\[ a = \frac{2r}{\sqrt{3}} \]

### Step 3: Calculate the perimeter of the hexagon.
The perimeter \( P \) of a regular hexagon is given by the formula:
\[ P = 6a \]
Substituting the expression for \( a \) we found in Step 2:
\[ P = 6 \left(\frac{2r}{\sqrt{3}}\right) = \frac{12r}{\sqrt{3}} \]

### Step 4: Simplify the expression for the perimeter.
To simplify \( \frac{12r}{\sqrt{3}} \), we can multiply the numerator and the denominator by \( \sqrt{3} \):
\[ P = \frac{12r \cdot \sqrt{3}}{3} = 4r\sqrt{3} \]

### Final Answer
Thus, the perimeter of the regular hexagon that circumscribes a circle of radius \( r \) is:
\[ P = 4r\sqrt{3} \]

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