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
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![**Title: Calculating the Area of the Shaded Region in a Circle**
**Problem Statement:**
Find the area of the shaded region in the figure. (Round your answer to two decimal places.)
**Diagram Explanation:**
The image shows a circle with a sector. The sector has a central angle of 120 degrees. Within the sector, there is a triangle formed by two radii and a chord. The length of the radius is given as 5 units.
**Steps to Solve:**
1. **Calculate the Area of the Sector:**
- The formula for the area of a sector with central angle \(\theta\) in degrees and radius \(r\) is:
\[
\text{Area of sector} = \frac{\theta}{360} \times \pi r^2
\]
- Here, \(\theta = 120^\circ\) and \(r = 5\).
2. **Calculate the Area of the Triangle:**
- The triangle within the sector is an isosceles triangle with two sides equal to the radius, and the included angle as the central angle (\(120^\circ\)).
- The area of the triangle can be calculated using the formula:
\[
\text{Area of triangle} = \frac{1}{2} r^2 \sin(\theta)
\]
- Here, use \(\theta = 120^\circ\) and \(r = 5\).
3. **Find the Area of the Shaded Region:**
- Subtract the area of the triangle from the area of the sector to find the area of the shaded region.
**Need Help?**
- Click on "Read It" for a detailed step-by-step explanation.
- Click on "Watch It" for a video tutorial.
Use these resources to guide you through the calculation process and deepen your understanding of geometric calculations involving sectors and triangles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e783f66-b34a-45d9-ad0a-1bd44c44ca06%2F87c1f0db-020a-4534-9cfc-34c9942e0d4c%2Fji9qois_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Area of the Shaded Region in a Circle**
**Problem Statement:**
Find the area of the shaded region in the figure. (Round your answer to two decimal places.)
**Diagram Explanation:**
The image shows a circle with a sector. The sector has a central angle of 120 degrees. Within the sector, there is a triangle formed by two radii and a chord. The length of the radius is given as 5 units.
**Steps to Solve:**
1. **Calculate the Area of the Sector:**
- The formula for the area of a sector with central angle \(\theta\) in degrees and radius \(r\) is:
\[
\text{Area of sector} = \frac{\theta}{360} \times \pi r^2
\]
- Here, \(\theta = 120^\circ\) and \(r = 5\).
2. **Calculate the Area of the Triangle:**
- The triangle within the sector is an isosceles triangle with two sides equal to the radius, and the included angle as the central angle (\(120^\circ\)).
- The area of the triangle can be calculated using the formula:
\[
\text{Area of triangle} = \frac{1}{2} r^2 \sin(\theta)
\]
- Here, use \(\theta = 120^\circ\) and \(r = 5\).
3. **Find the Area of the Shaded Region:**
- Subtract the area of the triangle from the area of the sector to find the area of the shaded region.
**Need Help?**
- Click on "Read It" for a detailed step-by-step explanation.
- Click on "Watch It" for a video tutorial.
Use these resources to guide you through the calculation process and deepen your understanding of geometric calculations involving sectors and triangles.
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