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?
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
