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In the diagram, two circles, each with center D, have radii of 1 and 2. The total area of the shaded region is 1/3 of the area of the larger circle. How many degrees are in the measure of (the smaller) angle ADC?

 

 Dec 4, 2020
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The area of the larger circle  will be  4 pi

 

So...the  shaded area  is  =  (1/3) * 4 pi  =  4/3*pi

 

Let  the measure  of  the smaller angle   ADC  = theta  (in rads)    

And let the measure of the larger angle ADC  = [2pi - theta ]  in rads

 

So...the  area of the sector subtended by the large circle  is (1/2) 4^2 * θ =  2θ

And let the area of the sector subtended by the larger circle be  (1/2) * 1^2 * (3pi - θ) =

(1/2) (3pi - θ)  

 

So we have that

 

2θ  +  (1/2)(3pi - θ)  =  (4/3)pi

 

θ = (4/7) pi  

 

So...the smaller angle ADC  =  (4/7) * 180 / pi  =  720/7*pi

 Dec 5, 2020

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