EXPLANATION:
Given;
We are given a decagon, which is a 10-sided polygon with center O, and a radius of 11 units.
Required;
We are required to find the area of the decagon and round the answer to the nearest tenth.
Step-by-step solution;
We begin by calculating the interior angle of e decagon.
We use the formula;
[tex]\begin{gathered} S=(n-2)180 \\ Where: \\ S=sum\text{ }of\text{ }interior\text{ }angles \\ n=number\text{ }of\text{ }sides \end{gathered}[/tex]
Therefore;
[tex]\begin{gathered} S=(10-2)180 \\ \\ S=8\times180 \\ \\ S=1440\degree \end{gathered}[/tex]
For a 10-sided polygon, each interior angle will now measure;
[tex]\begin{gathered} Interior\text{ }angle=\frac{1440}{10} \\ \\ Interior\text{ }angle=144\degree \end{gathered}[/tex]
take note that the radius of the polygon divides the angle at the vertex (that is, the interior angle) into two equal halves.
With that in mind we can now construct the following triangle;
Take note that the line from the center to the side is the apothem, while the side labeled 11 is the radius (which was given). We can now extract one of the right angled triangles and use that calculate the apothem as follows;
The side labeled a, the apothem can be calculated using the angle 72 degrees as the reference angle;
[tex]\begin{gathered} Reference\text{ }angle=72 \\ Opposite=a \\ Hypotenuse=11 \end{gathered}[/tex]
Note that;
[tex]sin\theta=\frac{opposite}{hypotenuse}[/tex][tex]sin72=\frac{a}{11}[/tex]
We cross multiply and we have;
[tex]\begin{gathered} 11\times sin72=a \\ \\ 10.46=a \end{gathered}[/tex]
Also, for the side labeled s (half the length of one side of the decagon);
[tex]cos\theta=\frac{adjacent}{hypotenuse}[/tex]
Therefore;
[tex]cos72=\frac{s}{11}[/tex]
We cross multiply and we now have;
[tex]\begin{gathered} 11\times cos72=s \\ \\ 3.39=s \end{gathered}[/tex]
For the length of one side of the decagon, we have the side s times 2. Hence, the length of the side is
[tex]\begin{gathered} Length\text{ }of\text{ }side=3.39\times2 \\ \\ Length\text{ }of\text{ }side=6.78 \end{gathered}[/tex]
The area of a polygon is given by the formula;
[tex]Area=\frac{1}{2}\times a\times p[/tex]
Where the variables are;
[tex]\begin{gathered} a=apothem \\ p=perimeter \end{gathered}[/tex][tex]\begin{gathered} Perimeter=6.78\times10 \\ Perimeter=67.8 \end{gathered}[/tex][tex]\begin{gathered} Area=\frac{1}{2}\times10.46\times67.8 \\ \\ Area=354.594 \end{gathered}[/tex]
Therefore, the area of the decagon rounded to the nearest tenth is;
ANSWER:
[tex]Area=354.594units^2[/tex]
