Question
A pedestrian crossing sign is the shape of a regular pentagon. The radius of the sign is 15 in. What is the apothem of the pentagon? Round your answer to 2 decimal places.
Answers
GPT-4o mini
To find the apothem of a regular pentagon, we can use the formula that relates the radius (\( R \)) and the apothem (\( a \)):
\[
a = R \cdot \cos\left(\frac{\pi}{n}\right)
\]
where \( n \) is the number of sides of the polygon. For a pentagon, \( n = 5 \).
Given:
- Radius \( R = 15 \) inches
- Number of sides \( n = 5 \)
Now, we calculate the angle:
\[
\frac{\pi}{n} = \frac{\pi}{5}
\]
Next, we compute \( \cos\left(\frac{\pi}{5}\right) \):
\[
\cos\left(\frac{\pi}{5}\right) \approx 0.8090
\]
Now, we can find the apothem:
\[
a = 15 \cdot \cos\left(\frac{\pi}{5}\right) \approx 15 \cdot 0.8090 \approx 12.135
\]
Finally, rounding to two decimal places, we have:
\[
a \approx 12.14 \text{ inches}
\]
Thus, the apothem of the pentagon is approximately **12.14 inches**.
\[
a = R \cdot \cos\left(\frac{\pi}{n}\right)
\]
where \( n \) is the number of sides of the polygon. For a pentagon, \( n = 5 \).
Given:
- Radius \( R = 15 \) inches
- Number of sides \( n = 5 \)
Now, we calculate the angle:
\[
\frac{\pi}{n} = \frac{\pi}{5}
\]
Next, we compute \( \cos\left(\frac{\pi}{5}\right) \):
\[
\cos\left(\frac{\pi}{5}\right) \approx 0.8090
\]
Now, we can find the apothem:
\[
a = 15 \cdot \cos\left(\frac{\pi}{5}\right) \approx 15 \cdot 0.8090 \approx 12.135
\]
Finally, rounding to two decimal places, we have:
\[
a \approx 12.14 \text{ inches}
\]
Thus, the apothem of the pentagon is approximately **12.14 inches**.