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
![### Calculating the Central Angle of a Sector
**Problem Statement:**
The area of the sector is given as \( 144 \pi \, \text{cm}^2 \). The radius of the sector, indicated as \( AF \) or \( AN \), is 36 cm. You are required to determine the measure of the central angle \( \angle FAN \) in degrees.
**Diagram Explanation:**
The diagram illustrates a sector \( AFN \) of a circle with the following details:
- The radius \( AF \) and \( AN \) is 36 cm.
- The area of the sector is \( 144 \pi \, \text{cm}^2 \).
- The angle at the center forming the sector is \( \angle FAN \), which needs to be calculated.
### Calculation:
To calculate the central angle (\( \angle FAN \)) of a sector, you can use the area formula for a sector:
\[ \text{Area of Sector} = \frac{\theta}{360} \times \pi r^2 \]
Where:
- \( \theta \) is the central angle in degrees.
- \( r \) is the radius of the circle.
Given:
- Area of Sector = \( 144 \pi \, \text{cm}^2 \)
- Radius, \( r \) = 36 cm
First, substitute the given values into the formula and solve for \( \theta \):
\[ 144 \pi = \frac{\theta}{360} \times \pi \times (36)^2 \]
Simplify the equation by dividing both sides by \( \pi \):
\[ 144 = \frac{\theta}{360} \times 1296 \]
Solve for \( \theta \):
\[ 144 = \frac{\theta \times 1296}{360} \]
\[ 144 \times 360 = \theta \times 1296 \]
\[ 51840 = \theta \times 1296 \]
\[ \theta = \frac{51840}{1296} \]
\[ \theta = 40 \]
Therefore, the measure of \( \angle FAN \) is 40 degrees.
**Answer:**
\[ m \angle FAN = 40 \, \text{degrees} \]
This completes the problem-solving process for determining the central angle of the sector.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f3257ee-1b98-40e5-a60e-ffe925108ca8%2F295e5088-3ebf-4f71-8531-c1d500eabc80%2F0vtm2k_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Central Angle of a Sector
**Problem Statement:**
The area of the sector is given as \( 144 \pi \, \text{cm}^2 \). The radius of the sector, indicated as \( AF \) or \( AN \), is 36 cm. You are required to determine the measure of the central angle \( \angle FAN \) in degrees.
**Diagram Explanation:**
The diagram illustrates a sector \( AFN \) of a circle with the following details:
- The radius \( AF \) and \( AN \) is 36 cm.
- The area of the sector is \( 144 \pi \, \text{cm}^2 \).
- The angle at the center forming the sector is \( \angle FAN \), which needs to be calculated.
### Calculation:
To calculate the central angle (\( \angle FAN \)) of a sector, you can use the area formula for a sector:
\[ \text{Area of Sector} = \frac{\theta}{360} \times \pi r^2 \]
Where:
- \( \theta \) is the central angle in degrees.
- \( r \) is the radius of the circle.
Given:
- Area of Sector = \( 144 \pi \, \text{cm}^2 \)
- Radius, \( r \) = 36 cm
First, substitute the given values into the formula and solve for \( \theta \):
\[ 144 \pi = \frac{\theta}{360} \times \pi \times (36)^2 \]
Simplify the equation by dividing both sides by \( \pi \):
\[ 144 = \frac{\theta}{360} \times 1296 \]
Solve for \( \theta \):
\[ 144 = \frac{\theta \times 1296}{360} \]
\[ 144 \times 360 = \theta \times 1296 \]
\[ 51840 = \theta \times 1296 \]
\[ \theta = \frac{51840}{1296} \]
\[ \theta = 40 \]
Therefore, the measure of \( \angle FAN \) is 40 degrees.
**Answer:**
\[ m \angle FAN = 40 \, \text{degrees} \]
This completes the problem-solving process for determining the central angle of the sector.
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