In a regular pentagon the side measures 6 cm and sum of the inradius and circumradius is 9cm. Ratio of inradius and circumradius is 1:2 respectively. Find the sum of the area of the circumcircle and the area of the incircle of the pentagon.

Options

A. `41 pi cm^2`
B. `45 pi cm^2`
C. `53 pi cm^2`
D. data insufficient

Answer

To solve the problem step by step, we will follow the information provided and the relationships between the inradius and circumradius of the regular pentagon.

### Step 1: Understand the given information
- The side length of the regular pentagon (s) = 6 cm.
- The sum of the inradius (r) and circumradius (R) = 9 cm.
- The ratio of inradius to circumradius is 1:2.

### Step 2: Set up the equations based on the ratio
Let the inradius be \( r \) and the circumradius be \( R \). From the ratio given, we can express:
- \( r = x \)
- \( R = 2x \)

### Step 3: Use the sum of inradius and circumradius
According to the problem, we have:
\[ r + R = 9 \]
Substituting the expressions for \( r \) and \( R \):
\[ x + 2x = 9 \]
This simplifies to:
\[ 3x = 9 \]
So,
\[ x = 3 \]

### Step 4: Find the values of inradius and circumradius
Now substituting back for \( r \) and \( R \):
- Inradius \( r = x = 3 \) cm
- Circumradius \( R = 2x = 2 \times 3 = 6 \) cm

### Step 5: Calculate the area of the incircle
The area \( A_{in} \) of the incircle is given by the formula:
\[ A_{in} = \pi r^2 \]
Substituting the value of \( r \):
\[ A_{in} = \pi (3)^2 = 9\pi \, \text{cm}^2 \]

### Step 6: Calculate the area of the circumcircle
The area \( A_{circ} \) of the circumcircle is given by the formula:
\[ A_{circ} = \pi R^2 \]
Substituting the value of \( R \):
\[ A_{circ} = \pi (6)^2 = 36\pi \, \text{cm}^2 \]

### Step 7: Find the sum of the areas
Now, we find the total area \( A_{total} \) which is the sum of the areas of the incircle and circumcircle:
\[ A_{total} = A_{in} + A_{circ} \]
Substituting the areas we calculated:
\[ A_{total} = 9\pi + 36\pi = 45\pi \, \text{cm}^2 \]

### Conclusion
Thus, the sum of the area of the circumcircle and the area of the incircle of the pentagon is:
\[ \boxed{45\pi \, \text{cm}^2} \]