Question
A pedestrian crossing sign is the shape of a regular pentagon. The radius of the sign is 15 in.
(1 point)
What is the apothem of the pentagon? Round your answer to 2 decimal places.
Question 2
What is the length of ONE side of the regular pentagon above? Round your answer to 2 decimal places.
(1 point)
Responses
17.63
17.63
8.82
8.82
88.17
88.17
12.14
12.14
Question 3
What is the area of the regular pentagon? Round your answer to the nearest whole number. (1 point)
in2
(1 point)
What is the apothem of the pentagon? Round your answer to 2 decimal places.
Question 2
What is the length of ONE side of the regular pentagon above? Round your answer to 2 decimal places.
(1 point)
Responses
17.63
17.63
8.82
8.82
88.17
88.17
12.14
12.14
Question 3
What is the area of the regular pentagon? Round your answer to the nearest whole number. (1 point)
in2
Answers
GPT-4o mini
To solve the problems regarding the regular pentagon, we can use the following formulas related to a regular pentagon with radius \( r \) (the circumradius), which is given as 15 inches.
### Question 1: Apothem of the Pentagon
The apothem \( a \) of a regular polygon can be calculated using the formula:
\[
a = r \cdot \cos\left(\frac{\pi}{n}\right)
\]
where \( n \) is the number of sides (for a pentagon, \( n = 5 \)).
Substituting \( r = 15 \) and \( n = 5 \):
\[
a = 15 \cdot \cos\left(\frac{\pi}{5}\right)
\]
Calculating \( \cos\left(\frac{\pi}{5}\right) \):
\[
\cos\left(\frac{\pi}{5}\right) \approx 0.8090
\]
Then,
\[
a \approx 15 \cdot 0.8090 \approx 12.14
\]
So the apothem is approximately **12.14 inches**.
### Question 2: Length of One Side of the Regular Pentagon
The length of one side \( s \) of a regular pentagon can be calculated using the formula:
\[
s = r \cdot 2 \cdot \sin\left(\frac{\pi}{n}\right)
\]
Substituting \( r = 15 \) and \( n = 5 \):
\[
s = 15 \cdot 2 \cdot \sin\left(\frac{\pi}{5}\right)
\]
Calculating \( \sin\left(\frac{\pi}{5}\right) \):
\[
\sin\left(\frac{\pi}{5}\right) \approx 0.5878
\]
Then,
\[
s \approx 15 \cdot 2 \cdot 0.5878 \approx 17.63
\]
So the length of one side is approximately **17.63 inches**.
### Question 3: Area of the Regular Pentagon
The area \( A \) of a regular polygon can be calculated using the formula:
\[
A = \frac{1}{2} \cdot Perimeter \cdot Apothem
\]
First, we find the perimeter:
\[
Perimeter = n \cdot s = 5 \cdot 17.63 \approx 88.15
\]
Now, using the apothem we found earlier:
\[
A \approx \frac{1}{2} \cdot 88.15 \cdot 12.14 \approx \frac{1}{2} \cdot 1070.89 \approx 535.44
\]
Rounding to the nearest whole number, the area is approximately **535 in²**.
### Summary of Answers:
1. Apothem: 12.14 inches
2. Length of one side: 17.63 inches
3. Area: 535 in²
### Question 1: Apothem of the Pentagon
The apothem \( a \) of a regular polygon can be calculated using the formula:
\[
a = r \cdot \cos\left(\frac{\pi}{n}\right)
\]
where \( n \) is the number of sides (for a pentagon, \( n = 5 \)).
Substituting \( r = 15 \) and \( n = 5 \):
\[
a = 15 \cdot \cos\left(\frac{\pi}{5}\right)
\]
Calculating \( \cos\left(\frac{\pi}{5}\right) \):
\[
\cos\left(\frac{\pi}{5}\right) \approx 0.8090
\]
Then,
\[
a \approx 15 \cdot 0.8090 \approx 12.14
\]
So the apothem is approximately **12.14 inches**.
### Question 2: Length of One Side of the Regular Pentagon
The length of one side \( s \) of a regular pentagon can be calculated using the formula:
\[
s = r \cdot 2 \cdot \sin\left(\frac{\pi}{n}\right)
\]
Substituting \( r = 15 \) and \( n = 5 \):
\[
s = 15 \cdot 2 \cdot \sin\left(\frac{\pi}{5}\right)
\]
Calculating \( \sin\left(\frac{\pi}{5}\right) \):
\[
\sin\left(\frac{\pi}{5}\right) \approx 0.5878
\]
Then,
\[
s \approx 15 \cdot 2 \cdot 0.5878 \approx 17.63
\]
So the length of one side is approximately **17.63 inches**.
### Question 3: Area of the Regular Pentagon
The area \( A \) of a regular polygon can be calculated using the formula:
\[
A = \frac{1}{2} \cdot Perimeter \cdot Apothem
\]
First, we find the perimeter:
\[
Perimeter = n \cdot s = 5 \cdot 17.63 \approx 88.15
\]
Now, using the apothem we found earlier:
\[
A \approx \frac{1}{2} \cdot 88.15 \cdot 12.14 \approx \frac{1}{2} \cdot 1070.89 \approx 535.44
\]
Rounding to the nearest whole number, the area is approximately **535 in²**.
### Summary of Answers:
1. Apothem: 12.14 inches
2. Length of one side: 17.63 inches
3. Area: 535 in²