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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).

Round your answer to two decimal places.


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