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# Hard geometry

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In the diagram, four circles of radius 4 units intersect at the origin. What is the number of square units in the area of the shaded region? Express your answer in terms of pi.

Note: The four centers are four vertices of a regular hexagon.

Apr 27, 2022

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Note: I'm not sure if this is right... pls check

Connecting the 4 dots gives us a trapezoid (outlined in green).

This trapezoid can be split into 3 equilateral triangles, each with an area $$4 \sqrt 3$$. Thus, the area of the trapezoid is $$12 \sqrt3$$

There are also 2 300 degree sectors, with a total area of $$\large{{300 \over 360} \times 16 \pi \times 2 = {80 \over 3} \pi}$$

We also need to subtract two small parts of the circle (colored in blue on the diagram).

The area of one of these sectors is a 60 degree sector - an equilateral triangle with side length 4. Thus, the area of one of these is: $${8 \over 3} \pi - 4 \sqrt 3$$

The area of the Venn diagram on top is $$16 \pi - {16 \over 3} \pi + 4\sqrt3$$ ($$16 \pi$$ is the area of the 2 semi-circles, and $$16- 4\sqrt3$$ is the area of the overlap.

Thus, the area of the shaded region is: $$12 \sqrt 3 + {80 \over 3} \pi - ({16 \over 3} \pi - 8 \sqrt 3) + 16 \pi - ({16 \over 3} \pi - 4 \sqrt 3)$$

This simplifies to: $$\color{brown}\boxed{32 \pi + 24\sqrt3}$$

Apr 27, 2022