If the perimeter of circle A is equal to perimeter of semi circle B, what is the ratio of their areas ?

Options

A. `(pi + 2)^(2): 2pi^(2)`
B. `2pi^(2) : (pi + 2)^(2)`
C. `(pi + 2)^(2) : 4pi^(2)`
D. `4pi^(2) : (pi + 2)^(2)`

Answer

To solve the problem, we need to find the ratio of the areas of circle A and semicircle B given that their perimeters are equal.

### Step 1: Define the Perimeter of Circle A
The perimeter (circumference) of a circle is given by the formula:
\[ P_A = 2\pi r_A \]
where \( r_A \) is the radius of circle A.

### Step 2: Define the Perimeter of Semicircle B
The perimeter of a semicircle includes the curved part and the diameter. The formula for the perimeter of a semicircle is:
\[ P_B = \pi r_B + 2r_B \]
where \( r_B \) is the radius of semicircle B. This simplifies to:
\[ P_B = \pi r_B + 2r_B = r_B(\pi + 2) \]

### Step 3: Set the Perimeters Equal
According to the problem, the perimeters are equal:
\[ 2\pi r_A = r_B(\pi + 2) \]

### Step 4: Solve for the Relationship Between Radii
From the equation above, we can express \( r_B \) in terms of \( r_A \):
\[ r_B = \frac{2\pi r_A}{\pi + 2} \]

### Step 5: Calculate the Areas
Now, we will calculate the areas of circle A and semicircle B.

- The area of circle A is given by:
\[ A_A = \pi r_A^2 \]

- The area of semicircle B is given by:
\[ A_B = \frac{1}{2} \pi r_B^2 \]

Substituting the expression for \( r_B \):
\[ A_B = \frac{1}{2} \pi \left( \frac{2\pi r_A}{\pi + 2} \right)^2 \]
\[ A_B = \frac{1}{2} \pi \cdot \frac{4\pi^2 r_A^2}{(\pi + 2)^2} \]
\[ A_B = \frac{2\pi^3 r_A^2}{(\pi + 2)^2} \]

### Step 6: Find the Ratio of the Areas
Now, we find the ratio of the areas \( \frac{A_A}{A_B} \):
\[ \frac{A_A}{A_B} = \frac{\pi r_A^2}{\frac{2\pi^3 r_A^2}{(\pi + 2)^2}} \]
\[ = \frac{(\pi + 2)^2}{2\pi^2} \]

### Final Step: Simplifying the Ratio
Thus, the ratio of the areas of circle A to semicircle B is:
\[ \frac{A_A}{A_B} = \frac{(\pi + 2)^2}{2\pi^2} \]

### Conclusion
The ratio of the areas of circle A to semicircle B is:
\[ \frac{(\pi + 2)^2}{2\pi^2} \]