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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I need to find the are of the sector if the radius is 8cm
![**Finding the Area of a Sector**
**Problem Statement:**
Find the area of the sector if the radius is 8 cm.
**Diagram Explanation:**
The provided diagram depicts a circle with center B. The circle contains a sector ∠ABC, where the angle at the center, AB, is marked as 51°. The radius of the circle, which is the distance from B to any point on the circle (such as A or C), is given as 8 cm.
**Step-by-Step Solution:**
1. **Identify Given Data:**
- Radius (r): 8 cm
- Central Angle (θ): 51°
2. **Formula for the Area of a Sector:**
The area of a sector is given by the formula:
\[
\text{Area} = \frac{\theta}{360} \times \pi r^2
\]
where:
- \( \theta \) is the central angle in degrees.
- \( r \) is the radius.
- \( \pi \) (pi) is a constant approximately equal to 3.14159.
3. **Substitute Values:**
Inserting the given values into the formula:
\[
\text{Area} = \frac{51}{360} \times \pi \times (8)^2
\]
4. **Calculate the Area:**
- Calculate \(8^2\):
\[
8^2 = 64
\]
- Multiply by \( \pi \):
\[
64\pi
\]
- Fraction of the circle ( \(\frac{51}{360}\) ):
\[
\frac{51}{360} \approx 0.1417
\]
- Finally, multiply all parts together:
\[
\text{Area} \approx 0.1417 \times 64\pi
\]
- Calculating it to two decimal places:
\[
\text{Area} \approx 0.1417 \times 201.062
\]
\[
\text{Area} \approx 28.5 \text{ cm}^2
\]
**Final Answer:**
The area of the sector is approximately 28.5 cm².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6756c5c2-608f-4015-b13f-deb910f8c3e1%2F0155dcd7-f14e-41ab-a7a2-f6f6cae2001d%2Fph2d9do_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Area of a Sector**
**Problem Statement:**
Find the area of the sector if the radius is 8 cm.
**Diagram Explanation:**
The provided diagram depicts a circle with center B. The circle contains a sector ∠ABC, where the angle at the center, AB, is marked as 51°. The radius of the circle, which is the distance from B to any point on the circle (such as A or C), is given as 8 cm.
**Step-by-Step Solution:**
1. **Identify Given Data:**
- Radius (r): 8 cm
- Central Angle (θ): 51°
2. **Formula for the Area of a Sector:**
The area of a sector is given by the formula:
\[
\text{Area} = \frac{\theta}{360} \times \pi r^2
\]
where:
- \( \theta \) is the central angle in degrees.
- \( r \) is the radius.
- \( \pi \) (pi) is a constant approximately equal to 3.14159.
3. **Substitute Values:**
Inserting the given values into the formula:
\[
\text{Area} = \frac{51}{360} \times \pi \times (8)^2
\]
4. **Calculate the Area:**
- Calculate \(8^2\):
\[
8^2 = 64
\]
- Multiply by \( \pi \):
\[
64\pi
\]
- Fraction of the circle ( \(\frac{51}{360}\) ):
\[
\frac{51}{360} \approx 0.1417
\]
- Finally, multiply all parts together:
\[
\text{Area} \approx 0.1417 \times 64\pi
\]
- Calculating it to two decimal places:
\[
\text{Area} \approx 0.1417 \times 201.062
\]
\[
\text{Area} \approx 28.5 \text{ cm}^2
\]
**Final Answer:**
The area of the sector is approximately 28.5 cm².
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