Question

a regular decagon has a radius of 8 cm. what is the approximate area of the decagon? recall that a decagon is a polygon with 10 sides 188 cm² 198 cm² 304 cm² 375 cm²

Explanation:

Step1: Calculate the central angle

The central angle of a regular decagon \(\theta=\frac{360^{\circ}}{n}\), where \(n = 10\). So \(\theta=\frac{360^{\circ}}{10}=36^{\circ}\)

Step2: Use the formula for the area of a regular polygon \(A=\frac{1}{2}nr^{2}\sin\theta\)

Here \(n = 10\), \(r = 8\mathrm{cm}\), \(\theta=36^{\circ}\)
Substitute the values into the formula:
\(A=\frac{1}{2}\times10\times8^{2}\times\sin36^{\circ}\)
First, calculate \(8^{2}=64\)
Then \(\frac{1}{2}\times10\times64 = 320\)
Since \(\sin36^{\circ}\approx0.618\)
\(A=320\times0.618 = 197.76\approx198\mathrm{cm}^2\)

Answer:

B. \(198\mathrm{cm}^2\)