Question

O is the center of the regular decagon below. Find its perimeter. Round to the nearesttenth if necessary.

Answer 1

Answer:

The perimeter of the decagon is 80.3

Step-by-step explanation:

Given:

radius = 13

polygon = decagon

To find:

The perimeter of the decagon

To determine the perimeter, we will apply the formula:

`Perimeter\text{ of a polygon = side }*\text{ number of sides}`

A decagon has 10 sides

number of sides = 10

We need to find the side length

Using an illustration:

Sum of angles at a point = 360°

Each central angle = 360/n = 360/10

Each central angle = 36°

s = side length, s/2 = half of the side length

To get the side length, we will apply sine ratio:

`\begin{gathered} sin36°\text{ = }(opposite)/(hypotenuse) \\ opposite\text{ = s/2, hyp = 13} \\ \\ sin\text{ 18 = }((s)/(2))/(13) \\ sin18\text{ = }(s)/(26) \\ s\text{ = 26sin 18} \\ \text{s = 8.03} \end{gathered}`

The perimeter of the decagon:

`\begin{gathered} Perimeter\text{ = 8.03 }*10 \\ Perimeter\text{ = 80.3 } \end{gathered}`