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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
 #1
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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

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