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:
[tex]\begin{gathered} \tan (54\degree)=\frac{6.2}{x} \\ x=\frac{6.2}{\tan(54\degree)} \\ x\approx4.5\text{ }mm \end{gathered}[/tex]
The length of the side of the pentagon is twice the length of x, that is,
[tex]\begin{gathered} \text{ Length of the side of the pentagon = }2x \\ \text{ Length of the side of the pentagon }\approx\text{ }2\cdot4.5 \\ \text{ Length of the side of the pentagon }\approx9\text{ }mm \end{gathered}[/tex]
Formula for the area of a regular pentagon
[tex]A=\frac{1}{4}\sqrt[]{5(5+2\sqrt[]{5})}\cdot a^2[/tex]
where a is the length of each side.
Substituting with a = 9 mm, we get:
[tex]\begin{gathered} A=\frac{1}{4}\sqrt[]{5(5+2\sqrt[]{5})}\cdot9^2 \\ A\approx139\text{ }\operatorname{mm}^2 \end{gathered}[/tex]
