\section*{Exercise 12 - V01012}

Draw the largest possible rhombus inside a regular hexagon with side length \(a\)!

a) Derive a formula for the area and perimeter of the rhombus!

b) What percentage of the hexagon's area does the rhombus occupy?
}

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1164.jpg?height=380&width=417&top_left_y=2214&top_left_x=317)

The area of the rhombus is: a) It is

\[
b=\sqrt{a^{2}+h^{2}}
\]

where \(h=\frac{\sqrt{3}}{2} a\), and therefore:

\[
b=\sqrt{a^{2}+\frac{3}{4} a^{2}}=\frac{\sqrt{7}}{2} a
\]

The perimeter of the rhombus is thus:

\[
U_{R}=4 b=2 \sqrt{7} a
\]

\[
A_{R}=4 \cdot \frac{1}{2} a h=2 a h=\sqrt{3} a^{2}
\]
b) The area of the hexagon is:

\[
A_{S}=6 \cdot \frac{1}{2} a h=3 a h
\]

The area of the rhombus is therefore two-thirds of the area of the hexagon, and thus approximately \(66.7 \%\).