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
![**Question:**
The radius of a circle is 6 meters. What is the length of a 180° arc?
**Diagram Explanation:**
The image illustrates a circle with a radius of 6 meters. A line from the center of the circle to the perimeter, labeled as "r = 6 m," represents the radius. The circle is divided into two equal parts by a diameter, creating two arcs. One of these arcs, highlighted in red, forms a 180° angle, covering half of the circle.
**Task:**
Calculate the exact length of the 180° arc and provide the answer in its simplest form.
**Answer Form:**
Provide your answer in the blank text box labeled "meters."
---
**Solution Explanation:**
The length of an arc (L) in a circle can be calculated using the formula:
\[ L = \theta \times r \]
where:
- \(\theta\) is the central angle in radians
- \(r\) is the radius of the circle
A 180° angle is equal to \(\pi\) radians (since 180° = \(\pi\) radians).
Given:
- Radius (r) = 6 meters
- \(\theta\) = \(\pi\) radians
Applying the values to the formula:
\[ L = \pi \times 6 = 6\pi \]
Therefore, the length of the 180° arc is \(6\pi\) meters.
**Answer:**
\[6\pi\] meters](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F84b65140-419d-48a3-803c-7ffe6d1f5ed0%2F86b5988a-e28e-47e9-8563-213aa8942f16%2Fghvmwl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
The radius of a circle is 6 meters. What is the length of a 180° arc?
**Diagram Explanation:**
The image illustrates a circle with a radius of 6 meters. A line from the center of the circle to the perimeter, labeled as "r = 6 m," represents the radius. The circle is divided into two equal parts by a diameter, creating two arcs. One of these arcs, highlighted in red, forms a 180° angle, covering half of the circle.
**Task:**
Calculate the exact length of the 180° arc and provide the answer in its simplest form.
**Answer Form:**
Provide your answer in the blank text box labeled "meters."
---
**Solution Explanation:**
The length of an arc (L) in a circle can be calculated using the formula:
\[ L = \theta \times r \]
where:
- \(\theta\) is the central angle in radians
- \(r\) is the radius of the circle
A 180° angle is equal to \(\pi\) radians (since 180° = \(\pi\) radians).
Given:
- Radius (r) = 6 meters
- \(\theta\) = \(\pi\) radians
Applying the values to the formula:
\[ L = \pi \times 6 = 6\pi \]
Therefore, the length of the 180° arc is \(6\pi\) meters.
**Answer:**
\[6\pi\] meters
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