Question
A circle with radius 16 centimeters is inscribed in a square.
que-4909175-1534235248208
Which is the area of the shaded region?
A
804.25 square feet
B
1024 square feet
C
822.94 square feet
que-4909175-1534235248208
Which is the area of the shaded region?
A
804.25 square feet
B
1024 square feet
C
822.94 square feet
Answers
GPT-4o mini
To find the area of the shaded region, we need to calculate the area of the square and then subtract the area of the inscribed circle.
1. **Calculate the area of the circle:**
- Radius \( r = 16 \) cm
- Area of the circle \( A_{\text{circle}} = \pi r^2 = \pi (16^2) = \pi (256) = 256\pi \).
2. **Calculate the area of the square:**
- The diameter of the circle is \( 2r = 32 \) cm, which is the same as the side length of the square.
- Area of the square \( A_{\text{square}} = \text{side}^2 = (32)^2 = 1024 \) square cm.
3. **Calculate the area of the shaded region:**
- Area of the shaded region \( A_{\text{shaded}} = A_{\text{square}} - A_{\text{circle}} \).
- This simplifies to \( A_{\text{shaded}} = 1024 - 256\pi \).
4. **Approximate the area of the shaded region:**
- Using \( \pi \approx 3.14 \):
- \( 256\pi \approx 256 \times 3.14 \approx 804.64 \) square cm.
- So, \( A_{\text{shaded}} \approx 1024 - 804.64 = 219.36 \) square cm.
Since your options are in square feet, we need to convert square centimeters to square feet:
- \( 1 \text{ cm}^2 = 0.00107639 \text{ ft}^2 \).
- Thus, we convert \( 219.36 \text{ cm}^2 \) to square feet:
- \( 219.36 \times 0.00107639 \approx 0.236 \text{ ft}^2 \).
None of the available options A, B, or C seem to match this calculation. Please check the values and options provided. If you're looking for the area of shaded space based on different dimensions or consideration, please provide that clarity. The inscribed circle and square dimensions provided do not lead to the provided answers.
1. **Calculate the area of the circle:**
- Radius \( r = 16 \) cm
- Area of the circle \( A_{\text{circle}} = \pi r^2 = \pi (16^2) = \pi (256) = 256\pi \).
2. **Calculate the area of the square:**
- The diameter of the circle is \( 2r = 32 \) cm, which is the same as the side length of the square.
- Area of the square \( A_{\text{square}} = \text{side}^2 = (32)^2 = 1024 \) square cm.
3. **Calculate the area of the shaded region:**
- Area of the shaded region \( A_{\text{shaded}} = A_{\text{square}} - A_{\text{circle}} \).
- This simplifies to \( A_{\text{shaded}} = 1024 - 256\pi \).
4. **Approximate the area of the shaded region:**
- Using \( \pi \approx 3.14 \):
- \( 256\pi \approx 256 \times 3.14 \approx 804.64 \) square cm.
- So, \( A_{\text{shaded}} \approx 1024 - 804.64 = 219.36 \) square cm.
Since your options are in square feet, we need to convert square centimeters to square feet:
- \( 1 \text{ cm}^2 = 0.00107639 \text{ ft}^2 \).
- Thus, we convert \( 219.36 \text{ cm}^2 \) to square feet:
- \( 219.36 \times 0.00107639 \approx 0.236 \text{ ft}^2 \).
None of the available options A, B, or C seem to match this calculation. Please check the values and options provided. If you're looking for the area of shaded space based on different dimensions or consideration, please provide that clarity. The inscribed circle and square dimensions provided do not lead to the provided answers.