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.
Answers
GPT-4o mini
To find the apothem of a regular pentagon when the radius (circumradius) is given, we can use the relationship between the circumradius \( R \) and the apothem \( a \).
The formula for the apothem \( a \) of a regular polygon with \( n \) sides and a circumradius \( R \) is given by:
\[
a = R \cdot \cos\left(\frac{\pi}{n}\right)
\]
For a regular pentagon, \( n = 5 \). Given that the radius \( R = 15 \) inches, we can plug in the values:
\[
a = 15 \cdot \cos\left(\frac{\pi}{5}\right)
\]
Next, we need to calculate \( \cos\left(\frac{\pi}{5}\right) \):
\[
\cos\left(\frac{\pi}{5}\right) \approx 0.8090
\]
Now, substituting this value into the equation for \( a \):
\[
a \approx 15 \cdot 0.8090 \approx 12.135
\]
Rounding this to two decimal places:
\[
a \approx 12.14
\]
Thus, the apothem of the pentagon is approximately **12.14 inches**.
The formula for the apothem \( a \) of a regular polygon with \( n \) sides and a circumradius \( R \) is given by:
\[
a = R \cdot \cos\left(\frac{\pi}{n}\right)
\]
For a regular pentagon, \( n = 5 \). Given that the radius \( R = 15 \) inches, we can plug in the values:
\[
a = 15 \cdot \cos\left(\frac{\pi}{5}\right)
\]
Next, we need to calculate \( \cos\left(\frac{\pi}{5}\right) \):
\[
\cos\left(\frac{\pi}{5}\right) \approx 0.8090
\]
Now, substituting this value into the equation for \( a \):
\[
a \approx 15 \cdot 0.8090 \approx 12.135
\]
Rounding this to two decimal places:
\[
a \approx 12.14
\]
Thus, the apothem of the pentagon is approximately **12.14 inches**.