A figure has a square of side 40 cm wih an circle in a circumcircle. The area of the region between the incircle and the circumcircle is (in `cm^(2)`)

Options

A. `100pi`
B. `200(pi - 2)`
C. `400pi`
D. `400(pi-1)`

Answer

To solve the problem, we need to find the area of the region between the incircle and the circumcircle of a square with a side length of 40 cm.

### Step-by-Step Solution:

1. **Identify the side length of the square**:
The side length of the square is given as 40 cm.

2. **Calculate the radius of the incircle**:
The radius of the incircle (r) of a square is half the side length.
\[ r = \frac{\text{side}}{2} = \frac{40}{2} = 20 \text{ cm} \]

3. **Calculate the diagonal of the square**:
The diagonal (d) of the square can be calculated using the formula:
\[ d = a\sqrt{2} = 40\sqrt{2} \text{ cm} \]

4. **Calculate the radius of the circumcircle**:
The radius (R) of the circumcircle is half the diagonal.
\[ R = \frac{d}{2} = \frac{40\sqrt{2}}{2} = 20\sqrt{2} \text{ cm} \]

5. **Calculate the area of the incircle**:
The area (A1) of the incircle is given by the formula:
\[ A1 = \pi r^2 = \pi (20)^2 = 400\pi \text{ cm}^2 \]

6. **Calculate the area of the circumcircle**:
The area (A2) of the circumcircle is given by the formula:
\[ A2 = \pi R^2 = \pi (20\sqrt{2})^2 = \pi (400 \cdot 2) = 800\pi \text{ cm}^2 \]

7. **Calculate the area of the region between the incircle and circumcircle**:
The area of the region between the incircle and circumcircle is:
\[ \text{Area} = A2 - A1 = 800\pi - 400\pi = 400\pi \text{ cm}^2 \]

### Final Answer:
The area of the region between the incircle and the circumcircle is \( 400\pi \text{ cm}^2 \).

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