Question
Find the area of a sector with a central angle of 194° and a radius of 10.2 millimeters. Round to the nearest tenth. 2 2 mm 69.1 a. c. 352.3 mm b. d. 176.1 mm2 17.3 mm2
Answers
GPT-4o mini
To find the area of a sector, you can use the formula:
\[
\text{Area} = \frac{\theta}{360} \times \pi r^2
\]
where:
- \(\theta\) is the central angle in degrees,
- \(r\) is the radius.
In your case, the central angle \(\theta\) is 194° and the radius \(r\) is 10.2 mm.
Let's calculate the area step by step:
1. **Calculate \(\pi r^2\)**:
\[
r^2 = (10.2 \, \text{mm})^2 = 104.04 \, \text{mm}^2
\]
\[
\pi r^2 \approx 3.14159 \times 104.04 \approx 326.725 \, \text{mm}^2
\]
2. **Calculate the fraction of the circle represented by the sector**:
\[
\frac{\theta}{360} = \frac{194}{360} \approx 0.53889
\]
3. **Calculate the area of the sector**:
\[
\text{Area} = 0.53889 \times 326.725 \approx 176.1 \, \text{mm}^2
\]
Thus, the area of the sector is approximately **176.1 mm²** when rounded to the nearest tenth.
The correct answer is option **d. 176.1 mm²**.
\[
\text{Area} = \frac{\theta}{360} \times \pi r^2
\]
where:
- \(\theta\) is the central angle in degrees,
- \(r\) is the radius.
In your case, the central angle \(\theta\) is 194° and the radius \(r\) is 10.2 mm.
Let's calculate the area step by step:
1. **Calculate \(\pi r^2\)**:
\[
r^2 = (10.2 \, \text{mm})^2 = 104.04 \, \text{mm}^2
\]
\[
\pi r^2 \approx 3.14159 \times 104.04 \approx 326.725 \, \text{mm}^2
\]
2. **Calculate the fraction of the circle represented by the sector**:
\[
\frac{\theta}{360} = \frac{194}{360} \approx 0.53889
\]
3. **Calculate the area of the sector**:
\[
\text{Area} = 0.53889 \times 326.725 \approx 176.1 \, \text{mm}^2
\]
Thus, the area of the sector is approximately **176.1 mm²** when rounded to the nearest tenth.
The correct answer is option **d. 176.1 mm²**.