To find the apothem, the length of one side, and the area of a regular pentagon with a radius of 15 inches, we can use some geometry formulas.
1. Apothem Calculation
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 inches), and \(n\) is the number of sides (5 for a pentagon).
Calculating: \[ a = 15 \cdot \cos\left(\frac{\pi}{5}\right) \]
Calculating \(\cos\left(\frac{\pi}{5}\right)\): \[ \cos\left(\frac{\pi}{5}\right) \approx 0.809 \]
So, \[ a \approx 15 \cdot 0.809 \approx 12.14 \text{ in} \]
2. Length of One Side Calculation
The length of one side \(s\) of a regular pentagon can be calculated using the formula: \[ s = r \cdot \sin\left(\frac{\pi}{n}\right) \cdot 2 \]
Calculating: \[ s = 15 \cdot \sin\left(\frac{\pi}{5}\right) \cdot 2 \]
Calculating \(\sin\left(\frac{\pi}{5}\right)\): \[ \sin\left(\frac{\pi}{5}\right) \approx 0.588 \]
So, \[ s \approx 15 \cdot 0.588 \cdot 2 \approx 17.64 \text{ in} \]
3. Area Calculation
The area \(A\) of a regular pentagon can be calculated using the formula: \[ A = \frac{1}{2} \times p \times a \] where \(p\) is the perimeter of the pentagon given by \(p = 5 \times s\).
We've found \(s \approx 17.64\), so: \[ p \approx 5 \cdot 17.64 \approx 88.20 \text{ in} \]
Now, we can find the area \(A\): \[ A \approx \frac{1}{2} \times 88.20 \times 12.14 \approx \frac{1}{2} \times 1072.99 \approx 536.50 \]
Rounding to the nearest whole number gives: \[ \text{Area} \approx 537 \text{ in}^2 \]
Summary of Answers
- Apothem: 12.14 in
- Length of One Side: 17.64 in
- Area: 537 in²
The responses for your quiz would therefore be:
- Apothem: 12.14 in
- Length of one side: 17.63 in (the closest option in your choices)
- Area: 537 in²
1 answer