Question

a regular decagon with center d is shown below. what is the area of the decagon? round your answer to the nearest tenth. square meters

Explanation:

Step1: Find the central angle

A decagon has \(n = 10\) sides. The central angle \(\theta=\frac{360^{\circ}}{n}=\frac{360^{\circ}}{10} = 36^{\circ}\)

Step2: Area formula for a regular polygon

The area formula for a regular polygon is \(A=\frac{1}{2}nr^{2}\sin\theta\), where \(r = 6\) m (radius), \(n = 10\), and \(\theta=36^{\circ}\)
Substitute the values into the formula:
\(A=\frac{1}{2}\times10\times6^{2}\times\sin(36^{\circ})\)
First, calculate \(6^{2}=36\)
Then \(\frac{1}{2}\times10\times36 = 180\)
We know that \(\sin(36^{\circ})\approx0.5878\)
\(A=180\times0.5878\)
\(A = 105.804\)

Answer:

\(105.8\) square meters