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?

Guest Dec 4, 2020

#1**0 **

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

Guest Dec 5, 2020