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

\(ABCD\) is a square of side length 1. Points \(E, F, G, H \) are such that \( AE : EB = 1:1\), \( BF: FC = 1 : 2 \), \( CG : GD = 1 : 3 \), \( DH : HA = 1 : 4 \).

What is the area of the quadrilateral \( EFGH \)?

In the figure above, we have a large blue circle of radius 1.

In this circle, we draw 2 equal red circles that are tangent to each other and to the large circle.

In each of these smaller red circles, we draw 3 equal blue circles in the same fashion.

In each of these smaller blue circles, we draw 4 equal red circles, and so on.

Compute the area of the red region.

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 \(\pi = \dfrac{22}{7}\).

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