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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 3

The side of the square is $$a$$. Express the green area with $$a$$.

Each square's vertex is a circumcentre of appropriate circular arc.

If the area of the shaded region equals $$7x - \cos x$$, what is the value of $$\frac \pi x ?$$

Two right triangles, both with legs measuring $$a$$ and $$b$$, where $$a>b$$, are overlapping as the image above.

What is the area of the gray quadrilateral?

###### Source: OBMEP 2016.

The front face of Captain America's shield (refer to this image drawn to scale) is made up of 4 circles and a 5-pointed regular star polygon which are all concentric. The 5-pointed star is tangential to the smallest circle at the 5 points seen in the figure. The diameters of the 4 circles are 30, 23, 18 and 12 inches respectively (yes, those are the real dimensions). Calculate the ratio $$\frac{\mbox{blue and red coloured area}}{\mbox{white area}}$$.

A circle of radius 3 units is centered at $$(-1, -1)$$. Find the area of the region that lies inside the circle and inside the first quadrant (See figure).