Geometry

Composite Figures

Composite Figures: Level 3 Challenges

         

The 2 green points in the diagram are the midpoints of 2 adjacent sides of a regular hexagon.

Which is larger, the red area or the blue area?

The diagram above shows that a semicircle is inscribed in a quarter circle while a small circle is inscribed in the semicircle. Given that \(AD=AE\) and the radius of the quarter circle is \(14\sqrt{2} \text{ cm}\), find the area of the green region above (in \(\text{cm}^{2}\)).

For your final step, use the approximation \(\pi = \dfrac{22}{7}\).

I begin with a circle of radius \(1 \text{ cm}\), which becomes the inscribed circle of an equilateral triangle, which I enclose with its circumcircle (the circle that touches all of its vertices). I enclose this circle with a square (so that each edge touches the circle once), and surround this with the square's circumcircle. I repeat this procedure with a pentagon, a hexagon, and so on forever (increasing the side number by one, until we reach an infinity-sided polygon).

What is the limiting area (in \(\text{cm}^2\)) of the "final" circle, rounded to a whole number?

\(ABCD\) is a square with points \(E\) and \(F\) lying on sides \(CD\) and \(AD,\) respectively. If the purple area is \([BHGI]=120,\) what is the sum of the pink areas \([AHF]+[FGED]+[ICE]?\)

What is the area of the large square?

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