If side of a square is 10 cm. Find ratio of area of circumcircle and In circle

Answer

To find the ratio of the area of the circumcircle to the area of the incircle of a square with a side length of 10 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the side length of the square**:
- Given side length \( a = 10 \) cm.

2. **Calculate the radius of the incircle**:
- The radius of the incircle (r) of a square is half the side length.
\[ r = \frac{a}{2} = \frac{10}{2} = 5 \text{ cm} \]

3. **Calculate the radius of the circumcircle**:
- The radius of the circumcircle (R) is half the length of the diagonal of the square. The diagonal \( d \) can be calculated using the formula \( d = a\sqrt{2} \).
\[ d = 10\sqrt{2} \text{ cm} \]
- Therefore, the radius of the circumcircle is:
\[ R = \frac{d}{2} = \frac{10\sqrt{2}}{2} = 5\sqrt{2} \text{ cm} \]

4. **Calculate the area of the incircle**:
- The area \( A_{in} \) of the incircle is given by the formula \( A = \pi r^2 \).
\[ A_{in} = \pi (5)^2 = 25\pi \text{ cm}^2 \]

5. **Calculate the area of the circumcircle**:
- The area \( A_{circ} \) of the circumcircle is given by the formula \( A = \pi R^2 \).
\[ A_{circ} = \pi (5\sqrt{2})^2 = \pi (25 \cdot 2) = 50\pi \text{ cm}^2 \]

6. **Find the ratio of the areas**:
- The ratio of the area of the circumcircle to the area of the incircle is:
\[ \text{Ratio} = \frac{A_{circ}}{A_{in}} = \frac{50\pi}{25\pi} = \frac{50}{25} = 2 \]
- Thus, the ratio of the area of the circumcircle to the area of the incircle is \( 2:1 \).

### Final Answer:
The ratio of the area of the circumcircle to the area of the incircle is \( 2:1 \).