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In the diagram, regular hexagon \(ABCDEF\) has sides of length  \(2\).

Using  \(A\), \(C\) ,\( E\) as centers, portions of circles with radius  \(1\) are drawn outside the hexagon.

Using \(B\),  \(D\) and \(F\) as centers, portions of circles with radius  \(1\) are drawn inside the hexagon. These six circular arcs join together to form a curve. Determine the area of the shaded region, both red and green, enclosed by this curve.

 

 Jan 14, 2021
 #1
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Area = (3√3 / 2 * a2) + pi

 Jan 14, 2021
 #2
avatar+114361 
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Red areas  =    3 ( area of  circular arc in each circle of 240° )  =  2* area of  circle with radius   =1 =   2pi

 

The white area is 1/2 of this   =   pi

 

Area of  hexagon  =    3sqrt (3)  (side^2)  / 2 =   3sqrt (3)    3sqrt (3) * 2^2  / 2 =   6sqrt (3)

 

Total of  red, green areas =  

 

2pi +  6sqrt (3) - pi  = 

 

pi + 6sqrt (3)  ≈    13.53

 

 

cool cool cool

 Jan 14, 2021

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