Question

a regular decagon is shown below. 8 yd. what is the area of the decagon? round your answer to the nearest tenth. square yards

Explanation:

Step1: Find the perimeter \(P\)

A decagon has \(n = 10\) sides. Given side length \(s=8\) yd.
Perimeter \(P=n\times s=10\times8 = 80\) yd.

Step2: Find the apothem \(a\)

The formula for the apothem of a regular polygon is \(a=\frac{s}{2\tan(\frac{180^{\circ}}{n})}\).
For \(n = 10\) and \(s = 8\), \(\frac{180^{\circ}}{n}=18^{\circ}\), \(\tan(18^{\circ})\approx0.3249\).
\(a=\frac{8}{2\times0.3249}=\frac{8}{0.6498}\approx12.31\) yd.

Step3: Calculate the area \(A\)

The formula for the area of a regular polygon is \(A=\frac{1}{2}Pa\).
Substitute \(P = 80\) and \(a\approx12.31\) into the formula:
\(A=\frac{1}{2}\times80\times12.31=40\times12.31 = 492.4\) square yards.

Answer:

\(492.4\)