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Overlap, inscribe, and circumscribe a collection of simple geometric shapes to make a complex, composite figure. See more

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Let \( ABCDE \) be a convex pentagon with \( \angle ABC= \angle BCD = 120^\circ \) and \([ABC]=[BCD]=[CDE]=[DEA]=[EAB]=12 \). Evaluate \([ABCDE] \) to 1 decimal place.

*Details:*

Notation: \( [ PQRS ] \) denotes the area of the figure \( PQRS \).

You may need to use a calculator to evaluate the area.

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In the figure, \(ABCD\) is a square of side length \(a\) units and \(E,F,G,H\) are the midpoints of the sides.

\(EAH, FBE,GCF,HDG\) are quadrants. There are two semicicles with diameters \(AD\) and \(BC\).

If the area of the shaded region in red can be expressed in the form

\[ \dfrac{a^2}x \left( \sqrt y - \dfrac\pi z\right) \]

then submit \( \dfrac {xy} z\) as your answer.

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A star is inscribed inside a circle with radius \(r\) as shown in the picture above. The formula that puts the area of the star \(S\) in terms of the circle's radius \(r\) can be expressed as follows:

\[ \large S = \dfrac{a \sqrt{10a-b\sqrt a}}c r^2 \; , \]

where \(a,b\) and \(c\) are positive integers with \(a\) square-free and \(a,c\) coprime.

Find the value of \(a+b+c\).

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Find the area of \(\color{violet}{Violet}\) region (four square looking region of central circle) if radius of each circle is 4 units.
###### Inspired by this question.

###### Image credit goes to Aniket Verma.

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