Question

1. 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.

2. What is the length of ONE side of the regular pentagon above? Round your answer to 2 decimal places.
(1 point)
12.14
8.82
17.63
88.17

Answer 1

1. To find the apothem \( a \) of a regular pentagon with a given radius \( r \) (here the radius is the distance from the center to a vertex), we can use the formula:

\[
a = r \cdot \cos\left(\frac{\pi}{n}\right)
\]

where \( n \) is the number of sides of the polygon (for a pentagon, \( n = 5 \)). The angle in the cosine function is \(\frac{\pi}{5}\).

Substituting in the values:

\[
a = 15 \cdot \cos\left(\frac{\pi}{5}\right)
\]

Calculating \(\cos\left(\frac{\pi}{5}\right)\):

\[
\cos\left(\frac{\pi}{5}\right) \approx 0.809
\]

Now, calculating \( a \):

\[
a \approx 15 \cdot 0.809 \approx 12.14 \text{ in}
\]

Rounding to two decimal places, the apothem of the pentagon is approximately:

\[
\boxed{12.14} \text{ in}
\]

---

2. To find the length of one side \( s \) of the regular pentagon, we can use 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 \(\sin\left(\frac{\pi}{5}\right)\):

\[
\sin\left(\frac{\pi}{5}\right) \approx 0.588
\]

Now, calculating \( s \):

\[
s = 15 \cdot 2 \cdot 0.588 \approx 15 \cdot 1.176 \approx 17.64 \text{ in}
\]

Rounding to two decimal places, the length of one side of the regular pentagon is approximately:

\[
\boxed{17.63} \text{ in}
\]

To sum up:
- The apothem is approximately \( 12.14 \) in.
- The length of one side is approximately \( 17.63 \) in.