Question
A figure is composed of a square and 4 semicircles. The square is 12in, find the area
Answers
GPT-4o mini
To find the area of the figure that consists of a square and 4 semicircles, we first need to calculate the area of the square and then the area of the semicircles.
1. **Area of the Square:**
The area \(A_{\text{square}}\) of a square is calculated using the formula:
\[
A_{\text{square}} = \text{side}^2
\]
Given that the side length of the square is 12 inches:
\[
A_{\text{square}} = 12^2 = 144 \text{ square inches}
\]
2. **Area of the 4 Semicircles:**
Each semicircle has a diameter equal to the side length of the square, which is 12 inches. Therefore, the radius \(r\) of each semicircle is half of the diameter:
\[
r = \frac{12}{2} = 6 \text{ inches}
\]
The area \(A_{\text{semicircle}}\) of one semicircle is given by the formula:
\[
A_{\text{semicircle}} = \frac{1}{2} \pi r^2
\]
Plugging in the radius:
\[
A_{\text{semicircle}} = \frac{1}{2} \pi (6^2) = \frac{1}{2} \pi (36) = 18\pi \text{ square inches}
\]
Since there are 4 semicircles, the total area \(A_{\text{4 semicircles}}\) is:
\[
A_{\text{4 semicircles}} = 4 \times A_{\text{semicircle}} = 4 \times 18\pi = 72\pi \text{ square inches}
\]
3. **Total Area of the Figure:**
The total area \(A_{\text{total}}\) of the figure is the sum of the area of the square and the total area of the 4 semicircles:
\[
A_{\text{total}} = A_{\text{square}} + A_{\text{4 semicircles}} = 144 + 72\pi
\]
Thus, the area of the figure composed of a square and 4 semicircles is:
\[
\boxed{144 + 72\pi} \text{ square inches}
\]
If you want a numerical approximation, substituting \(\pi \approx 3.14\):
\[
A_{\text{total}} \approx 144 + 72 \times 3.14 \approx 144 + 226.08 \approx 370.08 \text{ square inches}
\]
So, \(A_{\text{total}} \approx 370.08 \text{ square inches}\).
1. **Area of the Square:**
The area \(A_{\text{square}}\) of a square is calculated using the formula:
\[
A_{\text{square}} = \text{side}^2
\]
Given that the side length of the square is 12 inches:
\[
A_{\text{square}} = 12^2 = 144 \text{ square inches}
\]
2. **Area of the 4 Semicircles:**
Each semicircle has a diameter equal to the side length of the square, which is 12 inches. Therefore, the radius \(r\) of each semicircle is half of the diameter:
\[
r = \frac{12}{2} = 6 \text{ inches}
\]
The area \(A_{\text{semicircle}}\) of one semicircle is given by the formula:
\[
A_{\text{semicircle}} = \frac{1}{2} \pi r^2
\]
Plugging in the radius:
\[
A_{\text{semicircle}} = \frac{1}{2} \pi (6^2) = \frac{1}{2} \pi (36) = 18\pi \text{ square inches}
\]
Since there are 4 semicircles, the total area \(A_{\text{4 semicircles}}\) is:
\[
A_{\text{4 semicircles}} = 4 \times A_{\text{semicircle}} = 4 \times 18\pi = 72\pi \text{ square inches}
\]
3. **Total Area of the Figure:**
The total area \(A_{\text{total}}\) of the figure is the sum of the area of the square and the total area of the 4 semicircles:
\[
A_{\text{total}} = A_{\text{square}} + A_{\text{4 semicircles}} = 144 + 72\pi
\]
Thus, the area of the figure composed of a square and 4 semicircles is:
\[
\boxed{144 + 72\pi} \text{ square inches}
\]
If you want a numerical approximation, substituting \(\pi \approx 3.14\):
\[
A_{\text{total}} \approx 144 + 72 \times 3.14 \approx 144 + 226.08 \approx 370.08 \text{ square inches}
\]
So, \(A_{\text{total}} \approx 370.08 \text{ square inches}\).