Regular pentagons are inscribed in two circles of radius `5 `and `2` units respectively. The ratio of their areas is

Explanation

To find the ratio of the areas of two regular pentagons inscribed in circles of radius 5 units and 2 units respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Area Formula for a Regular Pentagon:**
The area \( A \) of a regular pentagon inscribed in a circle of radius \( r \) can be calculated using the formula:
\[ A = \frac{5}{2} r^2 \sin(72^\circ) \]
where \( r \) is the radius of the circumscribed circle.

2. **Calculate the Area of the First Pentagon:**
For the first pentagon inscribed in the circle of radius \( r_1 = 5 \):
\[ A_1 = \frac{5}{2} (5^2) \sin(72^\circ) = \frac{5}{2} \cdot 25 \cdot \sin(72^\circ) = \frac{125}{2} \sin(72^\circ) \]

3. **Calculate the Area of the Second Pentagon:**
For the second pentagon inscribed in the circle of radius \( r_2 = 2 \):
\[ A_2 = \frac{5}{2} (2^2) \sin(72^\circ) = \frac{5}{2} \cdot 4 \cdot \sin(72^\circ) = 10 \sin(72^\circ) \]

4. **Find the Ratio of the Areas:**
To find the ratio of the areas \( \frac{A_1}{A_2} \):
\[ \frac{A_1}{A_2} = \frac{\frac{125}{2} \sin(72^\circ)}{10 \sin(72^\circ)} \]
The \( \sin(72^\circ) \) cancels out:
\[ \frac{A_1}{A_2} = \frac{125/2}{10} = \frac{125}{20} = \frac{25}{4} \]

5. **Conclusion:**
The ratio of the areas of the two regular pentagons is:
\[ \frac{A_1}{A_2} = \frac{25}{4} \]