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.
What is the length of ONE side of the regular pentagon above? Round your answer to 2 decimal places. (1 point) Responses 8.82 8.82 88.17 88.17 17.63 17.63 12.14
What is the area of the regular pentagon? Round your answer to the nearest whole number. (1 point)
What is the length of ONE side of the regular pentagon above? Round your answer to 2 decimal places. (1 point) Responses 8.82 8.82 88.17 88.17 17.63 17.63 12.14
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 related to the regular pentagon, we'll need to use some geometric formulas.
1. **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 \( r \) is the radius (15 in) and \( n \) is the number of sides (5 for a pentagon).
So, we have:
\[
a = 15 \cdot \cos\left(\frac{\pi}{5}\right)
\]
First, calculate \( \frac{\pi}{5} \):
\[
\frac{\pi}{5} \approx 0.6283
\]
Then, calculate \( \cos\left(\frac{\pi}{5}\right) \):
\[
\cos(0.6283) \approx 0.8090
\]
Now plug it back into the apothem formula:
\[
a = 15 \cdot 0.8090 \approx 12.14 \text{ in}
\]
2. **Length of ONE side of the Regular Pentagon**:
The length of one side \( s \) of a regular pentagon can be calculated by the formula:
\[
s = 2 \cdot r \cdot \sin\left(\frac{\pi}{n}\right)
\]
So, plugging in:
\[
s = 2 \cdot 15 \cdot \sin\left(\frac{\pi}{5}\right)
\]
First, calculate \( \sin\left(\frac{\pi}{5}\right) \):
\[
\sin(0.6283) \approx 0.5878
\]
Now plug it in:
\[
s = 2 \cdot 15 \cdot 0.5878 \approx 17.63 \text{ in}
\]
3. **Area of the Regular Pentagon**:
The area \( A \) of a regular pentagon can be calculated using the formula:
\[
A = \frac{1}{2} \cdot Perimeter \cdot a
\]
The perimeter \( P \) is given by the number of sides times the length of one side:
\[
P = n \cdot s = 5 \cdot 17.63 \approx 88.17 \text{ in}
\]
Now calculate the area:
\[
A = \frac{1}{2} \cdot 88.17 \cdot 12.14 \approx 535.06 \text{ in}^2
\]
Rounding to the nearest whole number, the area is:
\[
A \approx 535 \text{ in}^2
\]
### Summary of Answers:
- **Apothem of the Pentagon**: 12.14 in
- **Length of ONE Side**: 17.63 in
- **Area of the Regular Pentagon**: 535 in²
1. **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 \( r \) is the radius (15 in) and \( n \) is the number of sides (5 for a pentagon).
So, we have:
\[
a = 15 \cdot \cos\left(\frac{\pi}{5}\right)
\]
First, calculate \( \frac{\pi}{5} \):
\[
\frac{\pi}{5} \approx 0.6283
\]
Then, calculate \( \cos\left(\frac{\pi}{5}\right) \):
\[
\cos(0.6283) \approx 0.8090
\]
Now plug it back into the apothem formula:
\[
a = 15 \cdot 0.8090 \approx 12.14 \text{ in}
\]
2. **Length of ONE side of the Regular Pentagon**:
The length of one side \( s \) of a regular pentagon can be calculated by the formula:
\[
s = 2 \cdot r \cdot \sin\left(\frac{\pi}{n}\right)
\]
So, plugging in:
\[
s = 2 \cdot 15 \cdot \sin\left(\frac{\pi}{5}\right)
\]
First, calculate \( \sin\left(\frac{\pi}{5}\right) \):
\[
\sin(0.6283) \approx 0.5878
\]
Now plug it in:
\[
s = 2 \cdot 15 \cdot 0.5878 \approx 17.63 \text{ in}
\]
3. **Area of the Regular Pentagon**:
The area \( A \) of a regular pentagon can be calculated using the formula:
\[
A = \frac{1}{2} \cdot Perimeter \cdot a
\]
The perimeter \( P \) is given by the number of sides times the length of one side:
\[
P = n \cdot s = 5 \cdot 17.63 \approx 88.17 \text{ in}
\]
Now calculate the area:
\[
A = \frac{1}{2} \cdot 88.17 \cdot 12.14 \approx 535.06 \text{ in}^2
\]
Rounding to the nearest whole number, the area is:
\[
A \approx 535 \text{ in}^2
\]
### Summary of Answers:
- **Apothem of the Pentagon**: 12.14 in
- **Length of ONE Side**: 17.63 in
- **Area of the Regular Pentagon**: 535 in²