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=AEAD=AE and the radius of the quarter circle is 142 cm14\sqrt{2} \text{ cm}, find the area of the green region above (in cm2\text{cm}^{2}).

For your final step, use the approximation π=227\pi = \dfrac{22}{7}.

I begin with a circle of radius 1 cm1 \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 cm2\text{cm}^2) of the "final" circle, rounded to a whole number?

ABCDABCD is a square with points EE and FF lying on sides CDCD and AD,AD, respectively. If the purple area is [BHGI]=120,[BHGI]=120, what is the sum of the pink areas [AHF]+[FGED]+[ICE]?[AHF]+[FGED]+[ICE]?

What is the area of the large square?

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