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
Find the Area of the regular Pentagon(5-gon) with a radius = 10 inches. Round your final answer to 2decimal places.
![### Problem 7: Area Calculation of a Regular Pentagon
**Question:**
Find the Area of the regular Pentagons (5-gon) with a radius \( r = 10 \) inches. Round your final answer to 2 decimal places.
---
**Diagram Explanation:**
- The image features a regular pentagon (a 5-sided polygon) with a highlighted radius extending from the center to one of the vertices.
- Within the pentagon, a right triangle is formed by dividing one of the triangular segments of the pentagon, indicating the height (or apothem) of the pentagon.
---
**Variables:**
- **Central Angle (\( \theta \)):**
The central angle of each triangular segment of the pentagon can be calculated as:
\[
\theta = \frac{360^\circ}{n} = \frac{360^\circ}{5} = 72^\circ
\]
- **Number of sides (\( n \)):**
\[
n = 5
\]
- **Side Length (\( s \)):**
The side length \( s \) can be calculated from the radius \( r \), using the sine function of half the central angle:
\[
s = 2 \times r \times \sin\left(\frac{\theta}{2}\right) = 2 \times 10 \times \sin(36^\circ) \approx 11.76 \text{ inches}
\]
- **Apothem (\( a \)):**
The apothem of the pentagon can be calculated using the cosine function of half the central angle:
\[
a = r \times \cos\left(\frac{\theta}{2}\right) = 10 \times \cos(36^\circ) \approx 8.09 \text{ inches}
\]
- **Area Calculation:**
The area \( A \) of a regular polygon can be calculated using the formula:
\[
A = \frac{1}{2} \times n \times s \times a
\]
Plugging in the values:
\[
A = \frac{1}{2} \times 5 \times 11.76 \times 8.09 \approx 237.76 \text{ square inches}
\]
---
**Answer:**
The area of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0f565ea-3faf-4daf-a447-59e0fa42aed0%2Fdbca5e9a-2e80-4863-a7f8-335e3a4929e6%2F5icno1l_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 7: Area Calculation of a Regular Pentagon
**Question:**
Find the Area of the regular Pentagons (5-gon) with a radius \( r = 10 \) inches. Round your final answer to 2 decimal places.
---
**Diagram Explanation:**
- The image features a regular pentagon (a 5-sided polygon) with a highlighted radius extending from the center to one of the vertices.
- Within the pentagon, a right triangle is formed by dividing one of the triangular segments of the pentagon, indicating the height (or apothem) of the pentagon.
---
**Variables:**
- **Central Angle (\( \theta \)):**
The central angle of each triangular segment of the pentagon can be calculated as:
\[
\theta = \frac{360^\circ}{n} = \frac{360^\circ}{5} = 72^\circ
\]
- **Number of sides (\( n \)):**
\[
n = 5
\]
- **Side Length (\( s \)):**
The side length \( s \) can be calculated from the radius \( r \), using the sine function of half the central angle:
\[
s = 2 \times r \times \sin\left(\frac{\theta}{2}\right) = 2 \times 10 \times \sin(36^\circ) \approx 11.76 \text{ inches}
\]
- **Apothem (\( a \)):**
The apothem of the pentagon can be calculated using the cosine function of half the central angle:
\[
a = r \times \cos\left(\frac{\theta}{2}\right) = 10 \times \cos(36^\circ) \approx 8.09 \text{ inches}
\]
- **Area Calculation:**
The area \( A \) of a regular polygon can be calculated using the formula:
\[
A = \frac{1}{2} \times n \times s \times a
\]
Plugging in the values:
\[
A = \frac{1}{2} \times 5 \times 11.76 \times 8.09 \approx 237.76 \text{ square inches}
\]
---
**Answer:**
The area of
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