1 Answer

Oamar Kanji · Answer 1 · 2023-03-19T02:50:37+0000

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:

$\begin{gathered} l=Ap\cdot2\tan \theta \\ \theta=(360)/(2\cdot n) \end{gathered}$

Find theta using the formula above, by replacing n for 10, which is the number of sides the decagon has.

$\begin{gathered} \theta=(360)/(2\cdot10) \\ \theta=(360)/(20) \\ \theta=18 \end{gathered}$

Now, find the length of the sides of the decagon.

$\begin{gathered} l=10\cdot2\cdot\tan 18 \\ l=6.5 \end{gathered}$

The length of each side is 6.5cm.

With this length, find the area of the decagon, use the following formula:

$A=(P\cdot Ap)/(2)$

Where P is the perimeter and Ap is the apothem.

Replace and find the area of the decagon:

$\begin{gathered} A=((6.5\cdot10)\cdot10)/(2) \\ A=325 \end{gathered}$

The area of the decagon is approximately 325 cm^2

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