A regular hexagon is inscribed in a circle of radius 5 cm. If `x` is the area inside the circle but outside the regular hexagon, then which one of the following is correct?

Options

A. `12 cm^(2) lt x lt 16 cm^(2)`
B. `15 cm^(2) lt x lt 17 cm^(2)`
C. `17 cm^(2) lt x lt 19 cm^(2)`
D. `19 cm^(2) lt x lt 21 cm^(2)`

Answer

To solve the problem of finding the area inside the circle but outside the regular hexagon, we will follow these steps:

### Step 1: Calculate the area of the circle
The formula for the area of a circle is given by:
\[ \text{Area of Circle} = \pi r^2 \]
where \( r \) is the radius of the circle. Given that the radius \( r = 5 \, \text{cm} \), we can substitute this value into the formula.

\[ \text{Area of Circle} = \pi (5)^2 = 25\pi \, \text{cm}^2 \]

### Step 2: Calculate the area of the regular hexagon
The formula for the area of a regular hexagon is given by:
\[ \text{Area of Hexagon} = \frac{3\sqrt{3}}{2} a^2 \]
where \( a \) is the length of a side of the hexagon. For a regular hexagon inscribed in a circle, the length of each side \( a \) can be calculated using the radius \( r \) of the circle:
\[ a = r = 5 \, \text{cm} \]

Now substituting \( a = 5 \) into the area formula for the hexagon:
\[ \text{Area of Hexagon} = \frac{3\sqrt{3}}{2} (5)^2 = \frac{3\sqrt{3}}{2} \cdot 25 = \frac{75\sqrt{3}}{2} \, \text{cm}^2 \]

### Step 3: Calculate the area inside the circle but outside the hexagon
To find the area \( x \) that is inside the circle but outside the hexagon, we subtract the area of the hexagon from the area of the circle:
\[ x = \text{Area of Circle} - \text{Area of Hexagon} \]
Substituting the values we calculated:
\[ x = 25\pi - \frac{75\sqrt{3}}{2} \]

### Step 4: Approximate the values
Now, we can approximate the numerical values:
- Using \( \pi \approx 3.14 \):
\[ 25\pi \approx 25 \times 3.14 = 78.5 \, \text{cm}^2 \]

- Using \( \sqrt{3} \approx 1.732 \):
\[ \frac{75\sqrt{3}}{2} \approx \frac{75 \times 1.732}{2} \approx \frac{129.9}{2} \approx 64.95 \, \text{cm}^2 \]

### Step 5: Final calculation of \( x \)
Now substituting these approximated values into the equation for \( x \):
\[ x \approx 78.5 - 64.95 \approx 13.55 \, \text{cm}^2 \]

### Conclusion
Thus, the area \( x \) inside the circle but outside the regular hexagon is approximately \( 13.55 \, \text{cm}^2 \).