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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.

 

 
 Nov 22, 2020
 #1
avatar+112375 
+1

Here's my best effort

 

(1/2) the area  of  one of the  larger "leaves" is

 

The area of a  sector with a 120° arc  and a radius of  4  less the area of a triangle  with sides of 4 and an included angle of 120° =   

 

(1/2) 4^2 (2pi/3)  -  (1/2)(4)^2 sin (120°)  = 

 

(1/2) 16  ( 2pi/3 - √3/2)  =

 

8  ( 2pi/3 - √3/2)

 

And we have 6 of these areas  (2 per leaf)

 

So the area  of all three larger leaves =  6 * 8  ( 2pi/3  - √3/2)  = 48 (2pi/3  - √3/2)  = 96 pi/3 - 24√3   (1)

 

From this....we must subtract  the total area  of the  smaller  two "leaves"

 

(1/2)  of the area of  one of these  = 

 

Area of a sector with a 60° arc and a radius of 4 less the area of a triangle with sides of 4 and an included angle of 60° = 

 

(1/2)  (4^2) (pi/3)  - (1/2)(4^2) sin 60°  =

 

(1/2)(4^2)  [ pi/3  - √3/2] =

 

8 [ pi/3  - √3/2]

 

And we have 4 of these areas

 

So the total area of the smaller two leaves =   32 [ pi/3 - √3/2]  =  32pi/3 - 16√3     (2)

 

Total shaded area  =  (1) - (2)  =    96pi/3 -24√3  - [ 32pi/3 - 16√3  ]  =

 

[64pi/3  - 8√3]  units^2

 

cool cool cool

 
 Nov 22, 2020

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