- 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:
\[ \boxed{12.14} \text{ in} \]
- 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.