Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Question
![**Finding Sector Area**
**Problem Statement:**
In circle G with \( m\angle FGH = 54^\circ \) and \( FG = 14 \) units, find the area of sector FGH. Round to the nearest hundredth.
**Diagram Explanation:**
- A circle is shown with center G.
- Points F and H lie on the circumference, forming the sector FGH.
- The angle ∠FGH is defined at 54 degrees.
- The length of segment FG is 14 units, indicating the radius of the circle.
**Solution:**
1. **Calculate the radius of the circle (r):**
Since segment FG is a radius of the circle, \( r = 14 \) units.
2. **Calculate the area of the entire circle:**
\[ \text{Area of circle} = \pi r^2 = \pi (14)^2 = 196\pi \]
3. **Determine the fraction of the circle occupied by the sector FGH:**
The angle in degrees is given as 54 degrees where a full circle has 360 degrees.
\[ \text{Fraction of the circle} = \frac{54}{360} = \frac{3}{20} \]
4. **Calculate the area of sector FGH:**
\[ \text{Area of sector FGH} = \left(\frac{3}{20}\right) \times 196\pi = \frac{588\pi}{20} = 29.4\pi \]
5. **Round to the nearest hundredth:**
Using \(\pi \approx 3.14159\),
\[ \text{Area of sector FGH} = 29.4 \times 3.14159 \approx 92.36 \]
Therefore, the area of sector FGH is approximately **92.36 square units**.
**Conclusion:**
We utilized the radius and the angle measure provided to determine the fraction of the circle's area represented by sector FGH and calculated the specific area, rounding it to the nearest hundredth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F16ab444c-cc67-4b2e-a5c5-3a02b2df29f8%2F677f8873-4556-466a-852d-5e96facd1f26%2Fhp37wuh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding Sector Area**
**Problem Statement:**
In circle G with \( m\angle FGH = 54^\circ \) and \( FG = 14 \) units, find the area of sector FGH. Round to the nearest hundredth.
**Diagram Explanation:**
- A circle is shown with center G.
- Points F and H lie on the circumference, forming the sector FGH.
- The angle ∠FGH is defined at 54 degrees.
- The length of segment FG is 14 units, indicating the radius of the circle.
**Solution:**
1. **Calculate the radius of the circle (r):**
Since segment FG is a radius of the circle, \( r = 14 \) units.
2. **Calculate the area of the entire circle:**
\[ \text{Area of circle} = \pi r^2 = \pi (14)^2 = 196\pi \]
3. **Determine the fraction of the circle occupied by the sector FGH:**
The angle in degrees is given as 54 degrees where a full circle has 360 degrees.
\[ \text{Fraction of the circle} = \frac{54}{360} = \frac{3}{20} \]
4. **Calculate the area of sector FGH:**
\[ \text{Area of sector FGH} = \left(\frac{3}{20}\right) \times 196\pi = \frac{588\pi}{20} = 29.4\pi \]
5. **Round to the nearest hundredth:**
Using \(\pi \approx 3.14159\),
\[ \text{Area of sector FGH} = 29.4 \times 3.14159 \approx 92.36 \]
Therefore, the area of sector FGH is approximately **92.36 square units**.
**Conclusion:**
We utilized the radius and the angle measure provided to determine the fraction of the circle's area represented by sector FGH and calculated the specific area, rounding it to the nearest hundredth.
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