Question
What is the area of the figure to the nearest hundredth?
Given
A square attached to half a circle
2 feet by 3 feet
Given
A square attached to half a circle
2 feet by 3 feet
Answers
GPT-4o mini
To find the area of the figure, we have a square and a half circle attached to it.
1. **Calculate the area of the square**:
- The side length of the square is 2 feet.
- Area of the square \( A_{\text{square}} = \text{side}^2 = 2 \, \text{ft} \times 2 \, \text{ft} = 4 \, \text{ft}^2 \).
2. **Calculate the area of the half circle**:
- The diameter of the half circle corresponds to the side of the square, which is 2 feet. Therefore, the radius \( r \) of the half circle is half of the diameter:
\[
r = \frac{2 \, \text{ft}}{2} = 1 \, \text{ft}.
\]
- The area of a full circle is given by the formula \( A_{\text{circle}} = \pi r^2 \).
- The area of a half circle will be:
\[
A_{\text{half circle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (1 \, \text{ft})^2 = \frac{1}{2} \pi \approx 1.5708 \, \text{ft}^2.
\]
3. **Sum the areas**:
- Now, we can find the total area of the figure by adding the area of the square and the area of the half circle:
\[
A_{\text{total}} = A_{\text{square}} + A_{\text{half circle}} = 4 \, \text{ft}^2 + \frac{1}{2} \pi \approx 4 + 1.5708 \approx 5.5708 \, \text{ft}^2.
\]
4. **Round to the nearest hundredth**:
- Rounding \( 5.5708 \, \text{ft}^2 \) to the nearest hundredth gives:
\[
A_{\text{total}} \approx 5.57 \, \text{ft}^2.
\]
Thus, the area of the figure is approximately \( \boxed{5.57} \) square feet.
1. **Calculate the area of the square**:
- The side length of the square is 2 feet.
- Area of the square \( A_{\text{square}} = \text{side}^2 = 2 \, \text{ft} \times 2 \, \text{ft} = 4 \, \text{ft}^2 \).
2. **Calculate the area of the half circle**:
- The diameter of the half circle corresponds to the side of the square, which is 2 feet. Therefore, the radius \( r \) of the half circle is half of the diameter:
\[
r = \frac{2 \, \text{ft}}{2} = 1 \, \text{ft}.
\]
- The area of a full circle is given by the formula \( A_{\text{circle}} = \pi r^2 \).
- The area of a half circle will be:
\[
A_{\text{half circle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (1 \, \text{ft})^2 = \frac{1}{2} \pi \approx 1.5708 \, \text{ft}^2.
\]
3. **Sum the areas**:
- Now, we can find the total area of the figure by adding the area of the square and the area of the half circle:
\[
A_{\text{total}} = A_{\text{square}} + A_{\text{half circle}} = 4 \, \text{ft}^2 + \frac{1}{2} \pi \approx 4 + 1.5708 \approx 5.5708 \, \text{ft}^2.
\]
4. **Round to the nearest hundredth**:
- Rounding \( 5.5708 \, \text{ft}^2 \) to the nearest hundredth gives:
\[
A_{\text{total}} \approx 5.57 \, \text{ft}^2.
\]
Thus, the area of the figure is approximately \( \boxed{5.57} \) square feet.