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Composite Figures

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

Level 4

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.

Notation: \( [ PQRS ] \) denotes the area of the figure \( PQRS \).
You may need to use a calculator to evaluate the area.

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.

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 \(a+b+c\).

The figure shows a unit circle with a square and an equilateral triangle inscribed in it.

The square and equilateral triangle share a vertex.

The overlapping area of the square and triangle can be expressed as \[a\sqrt{b}+\dfrac{c}{d}, \] where \(a,b,c\) and \(d\) are integers with \(b\) square-free and \(c,d\) coprime.

Find \(ab+cd\).

In a triangle \(ABC\), a random line passes through its centroid (the intersection of the three medians), segmenting it into two segments. Find the minimum ratio of the area of the smaller segment to the area of the larger segment.


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