Respuesta :
The first step to solve this problem is to find the length of the side of the polygon, in this case the decagon, using the following formulas:
[tex]\begin{gathered} l=Ap\cdot2\tan \theta \\ \theta=\frac{360}{2\cdot n} \end{gathered}[/tex]Find theta using the formula above, by replacing n for 10, which is the number of sides the decagon has.
[tex]\begin{gathered} \theta=\frac{360}{2\cdot10} \\ \theta=\frac{360}{20} \\ \theta=18 \end{gathered}[/tex]Now, find the length of the sides of the decagon.
[tex]\begin{gathered} l=10\cdot2\cdot\tan 18 \\ l=6.5 \end{gathered}[/tex]The length of each side is 6.5cm.
With this length, find the area of the decagon, use the following formula:
[tex]A=\frac{P\cdot Ap}{2}[/tex]Where P is the perimeter and Ap is the apothem.
Replace and find the area of the decagon:
[tex]\begin{gathered} A=\frac{(6.5\cdot10)\cdot10}{2} \\ A=325 \end{gathered}[/tex]The area of the decagon is approximately 325 cm^2
