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
12.14
12.14
8.82
8.82
88.17
88.17
Question 3
What is the area of the regular pentagon? Round your answer to the nearest whole number. (1 point)
(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
12.14
12.14
8.82
8.82
88.17
88.17
Question 3
What is the area of the regular pentagon? Round your answer to the nearest whole number. (1 point)
Answers
GPT-4o mini
To solve the problems concerning the regular pentagon, we need to use the properties of regular polygons. The given radius (R), which is the distance from the center of the pentagon to a vertex, is 15 inches.
### Question 1: Find the apothem of the pentagon.
The apothem (a) of a regular pentagon 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 the values:
\[
a = 15 \cdot \cos\left(\frac{\pi}{5}\right)
\]
Calculating \( \frac{\pi}{5} \):
\[
\cos\left(\frac{\pi}{5}\right) \approx 0.8090
\]
Now substituting this back into the apothem formula:
\[
a = 15 \cdot 0.8090 \approx 12.14
\]
Thus, the apothem of the pentagon is approximately **12.14 inches**.
### Question 2: Find the length of one side of the regular pentagon.
The length of one side (s) of a regular pentagon can be calculated using the formula:
\[
s = 2R \cdot \sin\left(\frac{\pi}{n}\right)
\]
Substituting the calculated values:
\[
s = 2 \cdot 15 \cdot \sin\left(\frac{\pi}{5}\right)
\]
Calculating \( \sin\left(\frac{\pi}{5}\right) \):
\[
\sin\left(\frac{\pi}{5}\right) \approx 0.5878
\]
Combining all values:
\[
s = 2 \cdot 15 \cdot 0.5878 \approx 17.63
\]
Thus, the length of one side of the pentagon is approximately **17.63 inches**.
### Question 3: Find the area of the regular pentagon.
The area (A) of a regular polygon can be calculated with the formula:
\[
A = \frac{1}{2} \times Perimeter \times Apothem
\]
The perimeter (P) of the pentagon is:
\[
P = 5s = 5 \cdot 17.63 \approx 88.15
\]
Now, substituting the apothem and perimeter into the area formula:
\[
A = \frac{1}{2} \times 88.15 \times 12.14
\]
Calculating the area:
\[
A \approx \frac{1}{2} \times 88.15 \times 12.14 \approx 535.76
\]
Rounding this to the nearest whole number gives us approximately **536 square inches**.
### Summary of Answers:
1. Apothem: **12.14 inches**
2. Length of one side: **17.63 inches**
3. Area: **536 square inches**
### Question 1: Find the apothem of the pentagon.
The apothem (a) of a regular pentagon 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 the values:
\[
a = 15 \cdot \cos\left(\frac{\pi}{5}\right)
\]
Calculating \( \frac{\pi}{5} \):
\[
\cos\left(\frac{\pi}{5}\right) \approx 0.8090
\]
Now substituting this back into the apothem formula:
\[
a = 15 \cdot 0.8090 \approx 12.14
\]
Thus, the apothem of the pentagon is approximately **12.14 inches**.
### Question 2: Find the length of one side of the regular pentagon.
The length of one side (s) of a regular pentagon can be calculated using the formula:
\[
s = 2R \cdot \sin\left(\frac{\pi}{n}\right)
\]
Substituting the calculated values:
\[
s = 2 \cdot 15 \cdot \sin\left(\frac{\pi}{5}\right)
\]
Calculating \( \sin\left(\frac{\pi}{5}\right) \):
\[
\sin\left(\frac{\pi}{5}\right) \approx 0.5878
\]
Combining all values:
\[
s = 2 \cdot 15 \cdot 0.5878 \approx 17.63
\]
Thus, the length of one side of the pentagon is approximately **17.63 inches**.
### Question 3: Find the area of the regular pentagon.
The area (A) of a regular polygon can be calculated with the formula:
\[
A = \frac{1}{2} \times Perimeter \times Apothem
\]
The perimeter (P) of the pentagon is:
\[
P = 5s = 5 \cdot 17.63 \approx 88.15
\]
Now, substituting the apothem and perimeter into the area formula:
\[
A = \frac{1}{2} \times 88.15 \times 12.14
\]
Calculating the area:
\[
A \approx \frac{1}{2} \times 88.15 \times 12.14 \approx 535.76
\]
Rounding this to the nearest whole number gives us approximately **536 square inches**.
### Summary of Answers:
1. Apothem: **12.14 inches**
2. Length of one side: **17.63 inches**
3. Area: **536 square inches**