13) We have a garden in the shape of a regular pentagon, if it is inscribed in a circle with a radius of 6 m and each side of the garden measures 7 m . What is the apothem of the garden? What is the area of the garden?

To find the apothem and the area of a regular pentagon inscribed in a circle, we can use the following formulas: 1. **Apothem (a)** of a regular polygon can be calculated using the formula: \[ a = \frac{s}{2 \tan(\frac{\pi}{n})} \] where \( s \) is the length of a side, and \( n \) is the number of sides. 2. **Area (A)** of a regular polygon can be calculated using the formula: \[ A = \frac{1}{2} \times Perimeter \times Apothem \] or \[ A = \frac{n \times s \times a}{2} \] ### Given: - Number of sides \( n = 5 \) (for a pentagon) - Length of each side \( s = 7 \) m ### Step 1: Calculate the Apothem Using the apothem formula: \[ a = \frac{s}{2 \tan(\frac{\pi}{n})} = \frac{7}{2 \tan(\frac{\pi}{5})} \] Calculating \( \tan(\frac{\pi}{5}) \): \[ \tan(\frac{\pi}{5}) \approx 0.7265 \] Now substituting this value into the apothem formula: \[ a = \frac{7}{2 \times 0.7265} \approx \frac{7}{1.453} \approx 4.81 \text{ m} \] ### Step 2: Calculate the Area The perimeter \( P \) of the pentagon is: \[ P = n \times s = 5 \times 7 = 35 \text{ m} \] Now using the area formula: \[ A = \frac{1}{2} \times P \times a = \frac{1}{2} \times 35 \times 4.81 \] Calculating the area: \[ A \approx \frac{1}{2} \times 35 \times 4.81 \approx 84.175 \text{ m}^2 \] ### Summary: - The apothem of the garden is approximately **4.81 m**. - The area of the garden is approximately **84.18 m²**.