The interior angles of a regular pentagon measure 108°.
The segment that passes through the center of a pentagon and one of the vertices bisects this angle.
Taking this information into consideration we can draw a right triangle inside the pentagon with the next measures:
With the help of the tangent function, the length of x is:

The length of the side of the pentagon is twice the length of x, that is,

Formula for the area of a regular pentagon
![A=(1)/(4)\sqrt[]{5(5+2\sqrt[]{5})}\cdot a^2](https://img.qammunity.org/2023/formulas/mathematics/college/8fmuxxeu2swxwwjkkmhg70rx65cckez743.png)
where a is the length of each side.
Substituting with a = 9 mm, we get:
![\begin{gathered} A=(1)/(4)\sqrt[]{5(5+2\sqrt[]{5})}\cdot9^2 \\ A\approx139\text{ }\operatorname{mm}^2 \end{gathered}]()